Distancia de un Punto a una Línea

Supongamos una recta y un punto en un plano cartesiano. ¿Qué se puede hacer para relacionar la recta y el punto y definir una característica que los correlacione?

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Distancia de un Punto a una Línea Distancia de un Punto a una Línea

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    Una forma fundamental de relacionarlos es la distancia entre ellos, y eso es lo que intentaremos averiguar en este artículo: la distancia de un punto a una recta.

    Definición de distancia de un punto a una recta

    Analicemos este problema mediante el siguiente diagrama.

    Distancia de un punto a una recta, una recta y un punto A con algunas distancias sobre él, StudySmarter

    Una recta l y un punto A con algunas de las distancias posibles entre ellos, StudySmarter Originals

    Si se nos pidiera hallar la distancia entre el punto A y la recta l , entonces sería realmente ambiguo, ya que hay varias, de hecho infinitas, formas en que podemos conectar el punto y la recta.

    Por tanto, hay que concretar sobre qué distancia se pregunta. Algunas de las distancias se muestran mediante los segmentos verdes en el diagrama anterior.

    Pero la única distancia específica que se puede nombrar y se puede cuantificar es la distancia más corta entre el punto y la recta.

    La distancia más corta entre la recta y el punto se muestra medianteel segmento de recta rosa en el diagrama anterior.

    La distancia entre un punto y una recta viene dada por la distancia más corta entre ellos.

    El segmento de recta más corto es perpendicular a la propia recta. Como cualquier otro segmento de recta formará un ángulo agudo u obtuso con la recta, la distancia más corta sólo será posible cuando sea perpendicular a la recta.

    Esta distancia puede verse alternativamente como el camino más corto por el que se puede llevar el punto A a la recta l.

    Pero, ¿cómo se puede determinar la distancia entre un punto y una recta utilizando la ecuación de una recta y las coordenadas del punto? Exploremos cómo podemos llegar a esa fórmula.

    Fórmula de la distancia de un punto a una recta

    Sea D una recta cuya ecuación viene dada por ax+by+c=0 donde a,b no son simultáneamente 0, y un punto A x0,y0 fuera de la recta, es decir, que no pertenece a la recta.

    El objetivo es encontrar la distancia más corta entre la recta D y el punto P. Sea Q el punto en el que el segmento de recta más corto interseca a la recta, cuyas coordenadas son x1,y1.

    La distancia entre el punto y la recta D es la misma que la longitud del segmento de recta formado por los puntos A y Q o la distancia entre ellos. Para ello podemos utilizar la fórmula de la distancia, pero para ello necesitamos conocer las coordenadas de Q en función de x0 and y0 para ello.

    Distancia de un punto a una recta, La distancia entre un punto y una recta, StudySmarterLa distancia entre un punto y una recta, StudySmarter Originals

    Recordemos que el gradiente de una recta con ecuación ax+by+c=0 viene dada por m1=-ab. Ahora el segmento de recta AQ es perpendicular a la recta, por lo que su pendiente será m2=ba. La razón es que el producto de las pendientes de dos rectas perpendiculares es siempre -1, es decirm1m2=-1.

    Ahora tenemos la pendiente de la recta que une AQ y las coordenadas de un punto A de la misma. Con esta información, podemos formar la ecuación de la recta AQ,

    y-y0=m2x-x0

    y-y0=bax-x0

    ay-ay0=bx-bx0

    Como Q se encuentra en esta recta, podemos sustituir x1,y1 por x,y para hallar las incógnitas x1 and y1.

    ay1-ay0=bx1-bx0

    Pero Q también está sobre la recta ax+by+c=0ax+by+c=0{"x":[[197,198,203],[213,214,215,219,219,219,216,215,213,212,205,202,200,190,175,169,167,165,163,160,158,157,155,155,155,156,159,162,167,169,179,182,193,196,208,212,215,222,225,230,237,238,244,246,249,251,254,257,260,261,267,269,275,277],[316,319,323,327,330,336,343,345,349,352,354,354,352,351,345,342,337,327,324,315,314,311,310],[395,395,396,396,396,396,392,390,388,383,380,377,372,369,367,365,362,360,358,358,358,360,361,366,369,378,382,394,402,406],[467,467,468,471,473,481,485,488,495,499,509,515,522,524,527,532,536,538,539],[492,492,492,493,493,494,494,494,494,494,493,493,493],[587,586,585,585,584,584,582,581,579,578,575,574,573,573,573,572,572,572,573,575,577,579,583,585,593,597,605,608,612,618,621,625,629,631,632,634,635,635,633,631,623,616,609,600,596,583,579,572,570],[685,686,686,687,687,687,686,685,685,685,684,684,687,689,690,694,696,698,701,706,708,710,716,719,721,726,729,733,737,740,741,742,743,743,743,742,742,742,741,741,741,741,741,740,739,738,736,735,733,730,726,724,717,714,706,703,696,693],[793,794,795,796,799,801,807,814,818,829,836,839,845,847,850,851,852,853],[827,827,827,826,827,829,831,832,833,834,834,834,833,833,833,833,834,835,836],[942,942,942,942,941,940,937,932,929,919,916,906,900,897,894,891,891,891,892,894,896,901,903,910,917,925,934,938],[982,984,989,997,1003,1009,1012,1021,1023,1027,1031,1032,1033],[975,975,976,978,980,982,986,989,993,999,1002,1008,1015,1020,1023,1029,1030],[1075,1076,1076,1076,1075,1074,1071,1069,1067,1066,1063,1062,1062,1062,1062,1062,1065,1068,1069,1073,1075,1077,1083,1085,1091,1096,1101,1104,1111,1113,1117,1118,1120,1120,1120,1120,1119,1118,1115,1112,1110,1108,1102,1098,1094,1088,1086,1081,1076,1073,1072]],"y":[[369,369,368],[345,344,342,326,324,319,311,309,307,306,304,304,304,309,330,342,347,354,360,373,380,385,398,402,406,407,408,406,404,402,393,390,378,373,361,357,354,349,347,346,346,348,358,362,373,379,391,403,412,416,424,426,429,428],[321,319,317,316,316,317,322,325,332,339,352,357,369,373,386,390,397,409,410,416,417,417,417],[330,329,327,323,322,320,320,321,323,328,331,335,343,348,353,358,368,374,387,394,401,408,411,418,419,420,420,417,413,409],[356,355,355,354,354,352,351,351,350,350,350,349,348,348,347,344,344,343,343],[327,327,329,334,341,345,360,364,376,379,389,391,394],[261,261,263,265,280,287,305,317,329,335,351,360,364,371,373,376,380,381,383,383,383,380,377,375,368,365,360,359,358,357,357,358,360,361,363,366,367,369,376,379,386,391,395,399,400,402,402,400,398],[327,327,326,328,332,339,347,355,363,366,376,378,382,383,384,384,384,383,382,379,377,374,368,365,362,355,352,345,339,334,333,331,331,334,336,345,350,353,363,373,386,398,405,418,424,430,441,446,456,464,470,473,478,479,478,477,470,467],[364,364,364,364,364,364,364,364,364,364,364,364,364,364,364,363,363,363],[337,336,335,333,331,332,334,337,344,353,368,373,383,393,398,403,405,410,411],[314,313,312,311,309,308,307,309,310,320,324,341,355,367,379,391,400,403,409,412,413,415,416,417,416,414,410,408],[339,339,339,338,337,336,335,334,334,333,333,333,334],[379,380,381,381,382,382,381,381,380,379,378,377,375,374,374,373,372],[307,305,304,303,303,306,314,323,335,343,365,372,377,389,393,397,404,411,413,417,419,420,422,422,421,417,412,408,395,390,374,368,350,344,338,327,322,317,308,304,301,299,296,295,295,297,299,303,307,311,315]],"t":[[0,20,36],[204,220,235,281,282,290,308,315,316,323,336,340,352,360,374,387,390,391,403,407,407,419,425,436,440,441,453,461,473,475,487,492,503,507,520,523,524,536,540,553,560,572,578,587,590,591,602,607,618,624,635,640,652,661],[1017,1026,1036,1043,1052,1057,1073,1074,1086,1090,1102,1107,1119,1124,1135,1140,1152,1157,1172,1174,1186,1190,1191],[1346,1351,1359,1373,1376,1387,1402,1407,1407,1419,1424,1424,1432,1440,1441,1452,1457,1467,1475,1486,1491,1502,1507,1519,1524,1535,1541,1552,1557,1569],[1754,1769,1776,1786,1791,1802,1807,1810,1819,1824,1836,1840,1852,1857,1857,1870,1878,1882,1887],[2043,2078,2082,2092,2102,2108,2119,2124,2136,2141,2152,2157,2158],[2392,2404,2419,2424,2436,2441,2449,2457,2469,2476,2489,2491,2504,2507,2508,2519,2524,2525,2536,2541,2553,2558,2569,2576,2589,2591,2604,2607,2608,2616,2624,2636,2641,2641,2653,2657,2658,2670,2677,2688,2691,2704,2708,2719,2724,2736,2741,2753,2757],[3088,3096,3104,3124,3135,3141,3153,3158,3169,3175,3190,3191,3204,3208,3208,3220,3224,3225,3237,3241,3241,3254,3258,3258,3270,3274,3275,3289,3291,3304,3308,3320,3325,3337,3341,3353,3358,3358,3366,3374,3389,3391,3404,3408,3408,3420,3425,3425,3437,3442,3453,3458,3470,3475,3483,3491,3504,3510],[3822,3831,3843,3853,3858,3858,3870,3875,3887,3894,3903,3911,3916,3925,3933,3933,3941,3953],[4064,4068,4078,4087,4102,4121,4125,4137,4141,4153,4158,4170,4175,4188,4195,4200,4208,4220,4225],[4451,4460,4462,4471,4476,4487,4494,4507,4508,4522,4525,4537,4541,4554,4558,4571,4575,4587,4592,4592,4604,4608,4609,4621,4625,4637,4642,4650],[4923,4930,4934,4935,4947,4950,4958,4970,4975,4987,4992,4992,5004],[5175,5184,5188,5189,5195,5203,5208,5209,5221,5225,5226,5238,5242,5254,5259,5271,52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lo que satisfará la ecuación de la recta D, por lo que tenemos

    ax1+by1+c=0ax1+by1+c=0{"x":[[203,204,205,205,206,207,207,208,208,208,208,205,204,202,201,197,195,192,187,184,181,174,171,166,161,156,154,149,148,146,146,148,150,153,155,157,160,165,168,171,177,180,183,190,192,198,203,208,212,216,218,222,222,225,225,226,226,227,227,227,227,227,228,229],[276,277,283,289,297,303,310,318,320,327,329,332,332,332,330,327,323,317,312,306,301,299,296],[351,349,346,343,340,337,335,329,326,323,321,314,312,308,308,307,308,310,314,319,322,331,335,339,347,351],[398,398,398,398,398,398,398,398,397,397,397,396,396,396,396],[439,440,443,446,447,449,457,460,469,472,475,478,483,490,494],[468,467,466,466,466,466,466,465,465,462,460,459,459,458,458,458,458,459],[550,550,550,550,550,550,549,548,547,546,543,543,541,540,540,539,539,538,538,539,540,542,545,547,551,553,555,560,566,571,579,584,588,590,592,592,589,587,580,578,569,567,564,559,556,554,552,549,548],[629,629,630,630,630,631,630,630,630,630,630,631,632,635,636,641,642,644,648,651,653,658,661,663,665,669,671,673,677,679,681,683,684,684,684,683,682,680,680,678,676,675,674,673,672,669,669,667,666,664,662,660,658,656,653,651,646,644],[711,711,711,712,712,712,713,714,715,715,715,716,716,717,717,717,717,717,719,720],[749,748,746,749,750,758,760,764,771,774,778,785,789,793,796,801,804,807,809,811],[771,773,774,775,776,776,777,777,775,774,771,770,769,767,766,765,765,764,764,764,764,765,766,768],[916,915,904,901,895,886,882,877,873,863,859,856,849,845,840,838,838,841,845,848,859,863,878,884,899,904],[965,968,970,974,982,985,991,994,997,1003,1006,1008,1011,1013,1014],[942,942,942,943,946,948,951,962,966,980,983,994,999,1002,1007,1010,1013,1016,1020],[1069,1069,1069,1068,1068,1065,1064,1063,1061,1059,1058,1057,1057,1057,1059,1063,1067,1068,1075,1078,1081,1087,1090,1099,1102,1104,1109,1114,1115,1119,1120,1122,1124,1124,1124,1123,1120,1115,1110,1105,1099,1092,1087,1081,1076,1070]],"y":[[263,262,261,260,260,257,255,249,247,240,237,231,228,227,226,224,224,224,226,227,230,236,240,248,258,270,277,295,302,313,316,322,324,325,325,325,325,322,320,317,312,309,305,298,294,287,281,275,271,269,268,270,272,281,285,289,297,302,306,309,317,320,324,333],[255,253,248,246,245,244,245,247,249,254,257,266,272,276,284,291,299,307,314,320,325,327,329],[253,254,254,256,260,262,266,273,278,282,287,301,305,318,322,328,331,336,340,343,344,346,346,346,345,344],[336,337,338,340,344,347,350,353,359,362,365,371,374,376,380],[284,285,286,287,287,288,289,289,289,289,289,288,287,286,285],[256,256,255,258,260,271,275,290,295,308,315,321,325,330,335,337,339,340],[203,207,211,221,233,240,247,258,263,268,279,283,290,294,297,302,304,309,311,312,311,310,308,307,304,303,302,300,298,296,296,296,296,298,301,302,308,310,316,318,324,326,327,330,331,331,331,331,330],[255,256,258,259,261,270,278,282,286,292,295,304,306,312,313,315,315,315,313,312,310,306,303,300,297,289,285,281,273,269,263,258,255,253,252,255,257,267,271,283,293,305,311,317,323,340,344,353,357,365,370,375,376,377,378,378,375,374],[335,336,337,339,340,341,346,349,352,355,360,363,366,374,379,384,387,390,391,390],[288,288,289,290,290,290,290,290,289,289,289,288,287,287,286,284,284,283,282,281],[252,252,252,252,252,253,259,262,274,279,292,297,301,309,312,316,319,325,328,330,334,336,338,339],[241,240,239,239,242,246,249,253,257,269,274,280,292,299,311,321,329,335,341,343,347,348,348,347,343,341],[272,272,272,272,272,272,271,271,270,270,269,269,269,269,269],[318,319,320,321,321,321,321,318,317,314,313,311,311,310,310,309,309,309,308],[232,233,237,243,247,264,270,281,293,306,311,317,327,331,338,344,349,351,354,355,355,355,353,347,344,341,334,325,320,310,306,300,289,284,274,264,256,247,239,232,227,224,222,222,223,230]],"t":[[0,7,11,19,27,39,45,56,62,72,77,90,94,94,106,111,111,123,127,128,139,144,144,156,162,173,177,189,195,206,211,223,227,240,244,244,256,261,261,273,277,278,290,294,294,306,312,322,328,340,344,356,361,373,377,378,390,394,398,406,411,411,422,430],[732,749,752,754,761,771,777,789,795,806,811,819,827,839,844,856,861,872,877,889,894,895,906],[1092,1108,1113,1114,1120,1128,1128,1139,1144,1145,1156,1161,1173,1178,1189,1194,1195,1206,1212,1223,1228,1239,1244,1247,1256,1261],[1554,1560,1570,1578,1589,1594,1595,1606,1611,1611,1623,1628,1628,1640,1644],[1964,1978,1981,1986,1989,1995,2006,2012,2023,2028,2028,2040,2045,2056,2061],[2240,2245,2249,2270,2278,2290,2296,2306,2312,2323,2328,2340,2345,2356,2361,2362,2373,2376],[2609,2637,2645,2657,2662,2662,2674,2678,2678,2690,2695,2695,2707,2712,2713,2724,2729,2740,2745,2757,2762,2774,2778,2790,2795,2795,2807,2812,2820,2828,2840,2846,2857,2862,2873,2879,2890,2895,2907,2912,2924,2928,2929,2937,2945,2948,2957,2962,2966],[3248,3261,3264,3265,3271,3278,3290,3295,3297,3307,3312,3324,3328,3340,3345,3357,3362,3362,3370,3379,3379,3390,3395,3396,3407,3412,3412,3424,3431,3431,3444,3445,3454,3463,3473,3490,3496,3507,3512,3524,3529,3540,3545,3546,3558,3564,3574,3579,3579,3591,3595,3607,3612,3613,3624,3629,3641,3646],[3943,3948,3953,3954,3962,3966,3974,3979,3979,3991,3995,3996,4008,4012,4024,4029,4041,4046,4058,4067],[4336,4351,4354,4371,4380,4390,4396,4396,4408,4412,4413,4425,4431,4432,4444,4445,4446,4454,4462,4474],[4634,4641,4647,4654,4655,4662,4674,4680,4691,4696,4708,4712,4713,4721,4729,4729,4741,4746,4746,4758,4762,4763,4775,4778],[5134,5150,5153,5155,5163,5173,5179,5180,5192,5196,5196,5208,5213,5213,5225,5230,5241,5246,5258,5263,5275,5279,5291,5296,5308,5312],[5605,5610,5615,5617,5629,5641,5646,5646,5659,5663,5663,5671,5679,5680,5688],[5826,5831,5843,5844,5854,5855,5863,5875,5880,5892,5896,5909,5913,5913,5926,5930,5930,5942,5946],[6193,6209,6213,6214,6222,6230,6238,6246,6258,6263,6263,6276,6280,6280,6292,6297,6309,6313,6325,6330,6332,6342,6347,6359,6363,6363,6376,6380,6392,6396,6397,6409,6413,6414,6426,6433,6445,6446,6455,6463,6475,6480,6492,6497,6509,6527]],"version":"2.0.0"}

    Las dos rectas anteriores se cruzan en Q y, por tanto, pueden resolverse simultáneamente para determinar las incógnitas x1 and y1escribiendo la primera ecuación en términos de x1,

    x1=-c-by1a

    Sustituyendo la expresión de x1 en ay1-ay0=bx1-bx0obtenemos

    ay1-ay0=b-c-by1a-bx0-bc-b2y1-a2y1+a2y0-abx0=0

    Resolviendo paray1 obtenemos

    ay1=ay0-bac+by1-bx0{"x":[[58,58,57,57,56,55,52,49,47,42,39,36,34,32,31,29,29,30,32,34,38,46,53,57,59,59,58,58,58,58,59,60,62,64,65,68,69,73,77],[89,89,89,89,89,90,91,91,93,94,96,103,108,112,118,122,123,124,124,123,121,120,120,119,119,118,118,118,117,115,114,113,111,108,106,105,102,100,95,93,92,88],[140,140,139,139,138,138,137,136,136,137],[177,176,175,176,177,182,189,191],[178,180,181,184,190,191,195,197,199],[301,301,301,301,300,296,291,289,286,282,280,279,278,278,279,283,287,289,294,296,300,301,302,302,302,302,302,303,305,308],[326,326,326,325,325,325,324,324,324,324,324,324,326,327,329,335,342,346,347,350,352,352,352,352,350,348,347,346,346,346,346,346,345,344,344,342,341,339,336,333,330,327,325],[367,367,367,367,368,369,369,369,368,368,368,368,368,370,371,375,380,381,382,383,382,379,377,372,370,369],[428,426,426,425,427,429,431,438,444,451,454],[515,515,516,516,516,516,516,516,515,514,513,513,512,512,510,510,511,513,516,518,519,522,525,526,528,529,529,525,520,516,512,511],[504,500,499,501,502,508,511,517,524,527,531,539,541,543],[517,519,519,519,519,517,517,514,507,499,495,494,494,495,498,501,503,509,514,519,520,521,522,522,522,523,526,528,529,531],[594,593,593,593,592,591,589,588,583,579,577,575,575,576,577,580,582,588],[627,627,626,625,623,620,616,614,614,616,618,622,630],[651,651,654,656,658,660,665],[662,661,660,659,659,659,659,659,659,659,659,659,659,659],[694,694,694,694,694,693,692,692,691,690,689,688,688,687,686,685,685,687,689,691,694,698,699,701,703,703,703,703,701,699,696,694,693,691],[727,727,727,728,728,728,728,728,728,727,726,726,729,734,738,743,750,751,753,753,753,752,751,750,748,747,745,744,742,741,740,740,739,739,738,736,732,731,729,726,725,722],[768,767,766,766,766,766,767,767,767,768,768,768,769],[802,802,802,803,804,805,808,809,813,813,814,812,810,809,804,802,796,794,791],[862,861,860,863,872,874,879,885,886,888,887,886,885,884],[928,928,927,927,926,926,926,925,925,924,923,922,921,919,917,916,916,918,923,926,928,933,934,938,939,939,939,937,933,928,925,924,923,922,920,919],[970,970,969,969,970,974,978,981,983,985,986,986,983,982,978,978,977],[1000,998,998,997,996,993,991,990,987,986,983,983,985,990,998,1005,1009],[1027,1027,1026,1025,1024,1024,1026,1027,1032,1035,1037,1038,1040,1038,1036,1034,1030,1028,1027]],"y":[[171,170,169,167,166,165,163,163,163,166,170,175,178,183,186,194,198,199,200,200,198,191,183,177,175,176,180,184,186,192,194,197,200,200,200,198,196,193,189],[172,169,168,167,168,171,173,175,186,190,193,192,189,185,178,172,170,165,164,164,166,170,172,178,181,187,190,195,209,218,227,230,236,243,244,245,245,245,242,241,239,234],[206,205,205,208,215,217,222,227,231,232],[183,183,182,182,181,181,181,181],[196,196,196,195,193,192,191,191,190],[181,178,177,176,173,172,173,174,178,184,186,190,192,193,196,196,194,193,190,189,185,185,184,185,186,187,188,190,192,192],[179,178,177,177,176,175,174,175,178,179,183,184,190,191,194,195,191,187,185,180,176,174,173,171,170,170,175,184,189,193,201,205,210,218,221,228,232,234,235,234,232,230,226],[226,225,224,223,222,222,223,226,230,231,234,236,238,240,240,238,232,230,227,223,222,220,220,219,220,220],[190,190,189,189,189,189,189,189,189,188,188],[163,158,157,156,155,154,157,158,163,170,173,175,178,180,188,192,193,191,189,187,186,185,185,185,186,188,191,194,196,197,197,196],[226,227,227,227,227,227,227,227,226,226,226,225,225,225],[268,267,266,265,263,261,260,260,261,268,273,278,279,281,281,281,280,277,274,271,271,271,274,275,278,281,282,282,281,280],[173,172,171,170,169,168,168,170,177,187,190,203,217,219,222,227,228,232],[184,182,181,180,180,184,190,196,199,200,201,201,199],[191,190,190,189,189,189,188],[184,183,183,182,183,184,185,186,188,189,192,197,198,199],[169,166,165,163,162,163,168,169,173,175,181,183,185,188,190,193,195,195,195,194,193,192,192,192,192,193,195,196,198,201,202,202,202,201],[183,182,181,179,178,177,178,180,182,187,194,199,202,202,201,199,195,194,192,191,190,189,189,189,190,190,192,193,196,197,201,209,213,224,231,236,241,242,242,239,238,233],[220,218,218,217,218,221,223,224,226,229,230,232,233],[163,162,161,161,163,164,170,172,182,186,195,207,210,214,220,223,229,231,234],[190,190,190,190,190,190,190,190,190,190,190,190,190,190],[166,162,161,159,159,158,159,159,164,165,172,175,183,191,197,198,200,199,195,193,192,191,191,191,191,194,196,200,203,206,207,208,208,208,206,206],[185,184,183,182,181,182,183,185,186,189,194,196,201,202,205,206,207],[183,181,180,181,183,187,189,190,194,196,202,204,208,210,210,210,209],[222,223,226,227,231,235,236,236,235,234,232,231,227,223,221,221,220,220,220]],"t":[[0,8,8,18,24,35,51,58,68,74,83,91,101,108,108,124,135,142,151,158,168,185,201,218,224,241,251,258,268,274,285,291,301,308,318,324,325,335,347],[858,859,868,884,901,908,923,925,943,950,959,975,985,991,1008,1018,1025,1035,1052,1068,1085,1091,1102,1108,1118,1125,1125,1135,1141,1152,1158,1171,1175,1185,1192,1194,1202,1209,1218,1225,1225,1235],[3034,3068,3075,3092,3109,3119,3126,3136,3153,3169],[3666,3676,3693,3717,3726,3736,3753,3761],[3905,3943,3943,3951,3959,3970,3976,3976,3987],[4466,4475,4479,4484,4503,4520,4536,4543,4553,4559,4570,4576,4576,4589,4593,4609,4620,4626,4639,4643,4654,4661,4670,4689,4693,4693,4704,4709,4726,4743],[4985,4993,5010,5010,5020,5026,5043,5060,5070,5076,5087,5093,5104,5110,5120,5137,5153,5160,5170,5176,5187,5193,5204,5210,5220,5237,5253,5269,5276,5287,5293,5293,5304,5310,5310,5324,5327,5337,5343,5352,5360,5370,5376],[5837,5841,5845,5852,5868,5894,5905,5910,5927,5927,5935,5946,5960,5971,5976,5987,6004,6010,6021,6027,6037,6043,6054,6060,6071,6075],[13649,13655,13658,13671,13689,13696,13707,13713,13730,13740,13757],[14263,14271,14276,14277,14281,14297,14313,14323,14330,14340,14346,14347,14358,14363,14374,14390,14407,14423,14430,14441,14446,14457,14463,14474,14480,14497,14507,14524,14538,14555,14563,14574],[14949,14952,14957,14977,14980,14990,14997,15007,15013,15014,15025,15030,15041,15047],[15343,15352,15356,15363,15380,15390,15397,15407,15424,15441,15457,15463,15472,15480,15491,15497,15509,15514,15530,15541,15547,15558,15574,15580,15591,15610,15624,15630,15631,15640],[16183,16189,16197,16208,16225,16230,16241,16247,16264,16275,16280,16291,16308,16314,16314,16325,16331,16342],[17445,17448,17464,17475,17492,17509,17524,17539,17557,17564,17578,17581,17598],[17792,17806,17839,17848,17860,17864,17881],[17997,17998,18006,18014,18049,18057,18064,18067,18076,18082,18092,18108,18115,18125],[18277,18280,18286,18290,18298,18331,18348,18351,18358,18367,18375,18381,18382,18393,18398,18409,18425,18442,18449,18463,18465,18481,18492,18498,18511,18516,18526,18531,18543,18559,18575,18581,18593,18598],[19017,19024,19032,19051,19065,19075,19101,19107,19115,19126,19142,19159,19176,19182,19193,19198,19215,19226,19232,19243,19248,19260,19265,19276,19282,19282,19294,19299,19310,19316,19326,19332,19343,19348,19360,19365,19382,19392,19398,19410,19415,19426],[19741,19756,19766,19773,19791,19798,19810,19815,19827,19832,19843,19848,19856],[20075,20081,20093,20107,20115,20126,20132,20143,20149,20162,20165,20177,20182,20182,20194,20199,20210,20215,20223],[20987,20988,21007,21032,21049,21060,21066,21077,21082,21094,21099,21110,21116,21121],[21312,21317,21325,21332,21344,21349,21360,21366,21377,21382,21394,21400,21411,21427,21432,21444,21449,21482,21499,21510,21516,21527,21532,21544,21550,21561,21566,21577,21593,21599,21610,21616,21616,21628,21632,21644],[22010,22015,22032,22049,22066,22083,22093,22099,22111,22116,22127,22133,22144,22150,22161,22166,22166],[22348,22355,22367,22391,22399,22410,22416,22416,22426,22433,22445,22449,22461,22477,22483,22494,22500],[22952,23025,23041,23050,23061,23077,23083,23095,23100,23111,23116,23128,23133,23150,23161,23167,23178,23183,23188]],"version":"2.0.0"}

    Expandiendo los paréntesis y reordenando los términos, obtenemos

    ay1=a2y0-bc-b2y1-abx0a

    Multiplicando ambos lados por aobtenemos

    a2y1+b2y1=a2y0-abx0-bc{"x":[[86,87,87,87,86,84,83,81,80,74,71,65,63,60,59,59,60,62,67,69,75,78,83,85,90,90,92,92,93,93,94,95,96,98,98,102,103,108,110],[115,111,110,112,113,117,120,121,123,123,124,125,126,126,124,122,118,114,114,117,125,130],[158,158,158,159,159,160,160,160,160,159,159,159,159,159,160,165,172,174,178,182,183,185,186,185,184,183,182,181,180,180,179,179,179,178,177,176,175,173,170,169,166,165,163,160],[204,204,204,204,205,205,205,205,205,204,203,203],[246,245,244,243,242,242,243,248,249,252,255,259,260,261],[256,255,255,255,255,254,252,251],[319,319,319,318,317,314,313,312,310,310,309,309,309,310,312,313,317,324,325,331,333,334,334,333,328,326,323,318,316,312],[348,349,349,349,349,349,351,352,352,355,356,356,354,351,349,348,346,350,352,360,363],[387,387,387,387,388,389,390,390,390,391,392,393,395,399,401,406,409,411,412,414,415,415,415,414,413,411,411,409,408,408,407,407,404,403,401,399,397,396,392,391],[441,440,440,440,440,441,441,441,442,442],[485,484,483,485,491,497,503,505,507],[487,486,485,486,487,491,495,499,505,511,512,515],[604,604,604,604,604,602,598,596,595,593,590,589,587,586,588,590,593,597,600,606,610,613,614,614,614,614,615],[627,627,627,627,627,627,627,629,632,635,636,636,634,633,631,630,627,626,625,627,637,642],[672,672,672,671,671,671,671,672,672,673,673,674,675,677,678,680,685,688,691,692,694,695,695,694,693,692,691,690,687,684,681,680,679,677,675,673,673],[704,704,704,705,707,708,708,709,709,709,709,709,709,709,710,711,712,714,716,717,717,717,717,709,708],[749,748,751,752,757,759,764,769,770,772,771],[811,812,813,813,814,814,814,815,815,812,810,807,803,800,799,797,794,794,795,798,799,802,804,807,808,811,813,813,814,815,815,816,817,822,823],[847,847,846,846,846,846,847,848,850,850,850,850,849,847,843,842,841,843,844,848,849,854,855,858,860,862,862,862,862,859,858,855,854,851,848,847,846],[892,891,890,890,890,889,892,896,899,899,900,900,900,899,898,894,893,895,901,903,908,909,913,914,915,914,912,909,909,908,908,908,910,913],[931,930,930,930,929,929,929,929,930,931,933,934,935,936,937,937,937,937,934,933,930,929],[970,968,967,968,974,975,978,983,985,987,988,990],[1025,1025,1025,1025,1025,1025,1025,1025,1025,1025,1024,1023,1021,1021,1018,1017,1017,1016,1017,1018,1021,1023,1027,1030,1032,1035,1036,1036,1036,1036,1035,1033,1032,1030,1027,1025,1023,1022,1021,1021],[1074,1075,1076,1076,1077,1078,1078,1078,1077,1074,1072,1069,1067,1065,1063,1064,1067,1071,1073,1080]],"y":[[219,218,216,214,210,208,208,208,208,212,215,223,227,235,239,242,242,243,242,241,236,234,228,226,220,219,217,218,218,221,222,228,235,238,239,241,241,238,236],[161,157,155,153,153,153,153,153,154,155,156,157,162,163,169,171,177,181,182,182,180,178],[202,200,199,197,195,195,197,199,202,209,211,213,217,218,222,223,219,218,215,211,209,205,200,199,198,198,199,201,205,214,225,228,235,245,249,251,255,257,260,260,260,259,258,255],[243,240,239,240,244,245,250,251,254,258,261,262],[219,219,219,219,219,220,221,223,223,223,223,223,223,222],[217,215,214,217,220,223,231,237],[199,196,195,194,197,209,215,219,229,231,237,239,242,243,243,242,241,238,238,237,238,240,241,244,247,248,249,250,250,250],[184,180,179,176,175,174,174,174,175,178,184,185,190,196,199,201,205,205,205,202,202],[220,218,217,216,215,216,219,221,224,230,235,236,237,237,236,233,229,226,224,221,216,215,214,213,213,219,222,228,235,238,241,251,263,265,270,270,269,268,265,263],[260,258,257,258,261,265,267,271,273,276],[224,224,224,224,223,223,223,223,223],[241,241,241,241,241,240,240,239,238,237,237,237],[237,236,234,232,230,229,231,232,234,236,242,244,249,252,252,251,248,245,242,238,236,237,241,242,244,246,247],[205,203,201,200,197,195,194,193,194,196,201,205,209,210,213,214,217,218,220,220,218,216],[225,223,222,221,220,221,222,226,232,233,235,237,238,239,239,239,236,234,231,230,226,224,223,225,231,235,241,243,252,261,269,271,273,274,272,271,270],[264,262,261,260,259,259,261,261,265,266,269,271,273,274,274,274,273,271,268,266,263,262,261,259,259],[232,232,232,232,231,231,231,231,231,231,232],[232,230,229,228,227,226,225,222,221,220,220,222,225,228,230,231,235,239,239,239,239,236,235,233,232,230,230,231,231,234,235,237,239,241,241],[204,203,202,200,199,198,198,199,203,205,211,216,218,227,236,237,240,239,239,237,236,234,233,232,232,233,234,235,237,241,242,244,244,245,245,244,242],[220,220,219,218,217,217,218,220,223,224,226,228,229,231,233,236,237,235,230,229,225,223,220,219,218,218,221,225,227,231,232,235,238,240],[250,250,251,252,253,254,257,258,260,260,260,260,260,258,257,255,254,253,251,251,250,249],[232,232,232,232,232,232,232,232,232,232,232,232],[210,207,206,205,203,202,203,204,209,212,220,223,231,234,241,244,245,246,246,246,243,242,240,239,239,239,240,241,242,244,245,247,248,249,250,250,250,250,248,247],[229,228,228,227,226,224,223,222,222,225,227,230,233,235,240,243,245,245,245,245]],"t":[[0,4,13,31,47,64,70,70,81,86,95,103,113,120,130,137,147,153,163,170,180,186,199,204,214,220,229,245,247,253,265,270,286,299,303,314,320,332,337],[604,613,629,646,654,664,675,6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a2+b2y1=a2y0-abx0-bc{"x":[[72,73,73,74,75,76,76,75,74,71,66,58,53,52,51,53,54,55,59,67,76,81,81,81,80,80,80,80,81,84,86],[108,105,104,103,103,104,105,109,113,115,115,115,114,110,106,105,104,107,109,115,118],[133,131,134,140,144,148,153,155],[148,147,146,145,145,145,144,143,142,142,143],[182,182,182,182,182,182,182,182,182,182,182,182,181,181,181,181,181,181,181,184,186,189,193,195,196,197,196,194,193,192,190,189,188,186],[212,211,210,210,209,209,210,211,212,213,216,219,221,222,222,219,218,215,215,213,215,217,219,221,227],[34,34,34,34,34,34,34,33,32,31,29,27,24,23,21,24,29,32],[251,251,251,252,255,256,258,260,261,264,264,264,259,255,250,245],[293,293,293,293,293,294,295,296,296,296,296,296,296,299,302,307,309,316,317,320,323,324,324,324,323,321,319,318,317,317,317,317,317,316,315,313,310,306,305],[361,361,361,360,360,360,358,357,357,357,357],[405,404,405,408,413,414,417],[402,400,399,401,407,409,413,416,424,427],[535,536,536,535,533,532,531,525,520,519,516,516,518,525,526,528,531,534,537,538,539,539,540,540,541,541,542,543,546,549,550],[562,561,560,559,559,559,559,561,565,569,570,570,568,565,563,561,560,560,564,572,576],[597,597,596,596,596,596,597,597,597,597,597,598,604,607,611,617,618,620,620,619,618,617,616,615,614,613,613,612,612,611,611,609,607,606,602,601,598,597],[638,638,638,638,639,639,638,638,638,639,640,642,646,647,648,649,650,649,648,646,641,640],[674,673,673,674,677,681,687,690,696],[748,748,748,748,747,745,743,740,739,734,731,730,729,729,730,732,736,739,740,743,745,746,749,750,751,753,754,754,755],[784,783,782,782,781,780,780,779,779,779,778,777,777,776,775,775,775,776,778,779,783,784,787,791,793,793,792,788,783,782,780,778,778],[820,820,819,819,818,819,822,824,825,826,826,827,826,825,823,822,821,822,823,824,826,830,832,838,843,844,843,843,841,840,839,838,838,838,841,842,843],[861,862,862,863,864,864,865,865,864,864,863,862,862,863,863,864,866,867,871,873,873,873,871,867,866,864],[897,896,899,901,903,908,910,917,918,922,923,924],[954,954,954,955,955,956,958,957,956,954,951,951,950,951,952,955,958,961,964,966,966,966,966,963,962,960,957,956,953,952,951],[998,998,998,998,998,998,996,995,994,990,989,986,985,983,983,985,986,990,996,999,1006,1008,1010]],"y":[[204,204,203,203,202,199,195,193,193,193,196,204,213,215,217,219,219,219,218,212,206,200,201,203,206,207,210,214,215,216,216],[165,164,163,162,161,160,160,159,159,162,163,164,169,174,178,180,180,180,179,177,177],[210,210,210,209,209,209,208,207],[201,200,199,198,199,200,206,212,219,220,225],[186,183,182,181,180,179,181,185,189,194,196,200,202,205,208,210,212,214,215,215,214,213,212,212,213,215,218,220,221,221,222,222,222,221],[163,161,161,160,159,157,157,155,155,154,153,153,154,158,160,165,167,172,173,177,178,178,177,177,176],[186,185,184,183,182,181,180,180,180,180,182,186,194,198,209,222,228,230],[173,170,169,169,171,172,174,179,182,191,194,200,210,217,223,228],[183,181,180,179,178,178,180,186,195,197,202,206,207,209,208,206,204,199,197,194,189,187,185,183,181,181,185,190,197,200,208,211,217,224,226,229,229,226,225],[225,223,222,223,224,228,235,238,240,243,244],[193,193,193,193,193,193,193],[210,211,212,212,212,211,210,210,208,208],[200,198,197,195,193,193,193,194,199,201,205,210,212,210,209,208,206,203,201,201,201,202,205,206,209,210,211,212,213,213,212],[168,166,166,165,164,163,162,160,159,160,163,167,172,175,179,182,183,184,184,181,180],[191,190,190,189,188,189,190,193,194,199,202,206,207,206,203,198,197,195,194,194,194,196,197,201,203,209,219,227,231,233,235,238,238,237,235,234,230,226],[225,224,223,222,223,225,229,230,233,235,235,234,231,230,229,227,226,223,222,221,221,221],[201,201,200,200,200,199,199,199,199],[209,206,205,204,202,201,201,201,202,206,210,214,215,216,217,217,216,214,213,212,212,211,211,211,211,212,214,215,216],[186,181,180,179,177,177,176,177,183,185,190,197,199,207,213,214,216,216,215,214,212,211,209,208,207,209,212,216,218,218,218,217,216],[203,202,202,201,200,201,203,204,206,206,208,211,212,214,217,217,218,216,214,213,212,207,205,201,197,197,199,200,203,205,209,210,213,216,218,218,218],[226,226,225,225,224,223,222,223,226,227,229,232,233,235,236,236,236,235,233,230,227,226,224,222,222,222],[203,203,204,204,204,204,204,203,203,203,203,203],[188,186,185,183,182,181,187,194,199,206,213,215,216,216,214,212,209,207,207,207,208,209,211,214,215,216,217,218,218,218,217],[204,201,200,199,198,196,196,196,196,198,199,203,205,210,211,215,216,217,218,218,217,217,216]],"t":[[0,6,21,29,39,54,71,89,96,108,112,129,146,157,163,172,179,179,189,206,223,239,263,273,279,289,297,305,313,323,339],[693,702,713,730,740,756,763,773,790,807,813,813,823,840,846,856,863,879,894,896,907],[1202,1216,1238,1256,1263,1273,1290,1294],[1424,1436,1439,1446,1480,1480,1496,1517,1521,1531,1540],[1684,1688,1697,1699,1707,1723,1738,1747,1757,1763,1774,1780,1780,1790,1796,1797,1807,1813,1824,1840,1846,1857,1874,1880,1889,1896,1913,1923,1930,1941,1946,1957,1963,1971],[2174,2179,2182,2189,2197,2213,2224,2230,2230,2241,2247,2257,2274,2293,2297,2307,2314,2326,2331,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    Ahora dividiremos por a2+b2{"x":[[213,214,214,214,214,214,214,214,212,210,208,206,200,198,191,189,183,179,176,175,174,174,174,175,176,179,181,182,204,207,215,218,227,231,232,234,238,241,243,247,249,251,256,258,260,266,269,275,280],[288,286,283,282,281,281,280,279,279,280,281,285,289,293,295,297,300,302,307,309,309,310,310,309,308,305,302,299,296,294,293,292,292,295,298,303,306,316,319,323,331],[374,375,376,378,379,387,391,402,409,414,421,424,427,436,442],[408,409,412,413,414,417,419,421,421,421,421,421,420,420,420],[503,503,503,503,504,505,505,505,505,505,505,504,504,503,503,503,504,504,505,506,508,510,511,513,523,527,531,535,539,542,543,544,543,542,537,534,529,527,523,517,516,514],[553,553,553,553,554,556,559,561,563,570,575,579,583,584,588,589,590,590,588,585,581,579,577,575,572,571,570,570,571,575,577,584,593,598,609,615]],"y":[[303,299,297,294,291,289,286,284,281,279,277,277,278,279,287,290,303,313,324,329,334,342,344,346,347,348,348,348,332,328,318,314,306,306,308,310,319,326,331,337,340,343,346,347,348,348,348,346,342],[220,218,216,215,214,213,210,207,205,202,200,197,195,193,193,193,193,194,198,202,205,212,215,221,225,232,238,245,250,255,257,262,263,265,265,265,265,263,263,262,261],[317,317,317,317,317,316,315,313,312,311,310,310,309,308,307],[294,293,294,295,297,302,309,317,326,330,337,341,344,349,354],[216,217,221,224,239,251,262,272,283,292,296,310,313,324,327,333,334,335,336,336,334,333,332,331,327,327,327,328,329,330,332,333,337,338,343,346,349,350,351,351,350,347],[200,198,195,194,191,190,188,186,186,184,184,184,186,187,190,191,195,196,201,206,210,212,214,217,220,222,223,225,226,226,226,225,223,222,220,218]],"t":[[0,8,13,15,21,30,35,35,46,52,63,68,80,85,96,102,113,118,130,135,136,146,152,152,163,169,169,180,213,218,230,235,257,260,261,271,280,285,296,302,303,313,318,319,330,335,336,344,352],[618,623,628,635,635,646,652,662,669,679,685,696,702,713,719,719,729,735,749,752,761,769,779,785,786,796,802,813,819,829,835,846,852,863,869,879,885,896,902,902,916],[1150,1161,1164,1165,1170,1179,1186,1196,1203,1213,1219,1219,1230,1235,1257],[1431,1442,1461,1469,1469,1480,1486,1494,1502,1513,1519,1519,1530,1536,1546],[1782,1788,1803,1812,1819,1830,1836,1845,1852,1863,1869,1883,1886,1896,1902,1913,1919,1919,1930,1936,1949,1952,1953,1961,1983,1986,1996,2003,2013,2019,2030,2037,2047,2052,2063,2069,2080,2086,2097,2102,2113,2119],[2376,2383,2390,2403,2403,2413,2419,2420,2430,2436,2447,2453,2463,2469,2480,2486,2497,2503,2513,2519,2530,2536,2536,2547,2553,2553,2564,2569,2580,2586,2586,2597,2603,2614,2619,2624]],"version":"2.0.0"}para obtener

    y1=a2y0-abx0-bca2+b2

    Sustituyendo esto de nuevo en x1=-c-by1a para determinar x1obtenemos

    x1=-ca-baa2y0-abx0-bca2+b2{"x":[[39,38,37,36,35,33,34,39,43,45,51,56,54,49,43,36,34,35],[83,82,81,81,79,75,74,70,67,64,62,60,58,61,70,76],[107,107,107,107,107,107,107,107,107,106,105,105],[146,145,145,144,146,147,151,158,160,162,164],[151,150,152,153,159,166,169,173],[238,237,236,235,236,238,240,244,246,252,257,259,260],[297,297,297,296,296,295,293,292,285,280,276,278,280,282,286,288,295],[270,268,267,266,265,266,269,274,289,293,300,309,311,313,311,309],[299,300,300,300,300,300,299,297,294,292,289,288,284,282,282,282,284,285,289,294,296,300,301,303,305,306,306,306,307,309],[347,346,345,344,343,344,349,358,369,372],[418,418,418,418,417,417,416,416,415,414,413,413,413,415,419,420,423,424,427,429,430,431,431,430,429,426,425,422,419,418,415,415],[406,405,406,407,411,416,427,439,446,446,445],[428,430,430,430,431,431,430,429,427,424,423,419,416,415,414,413,413,413,415,420,421,425,426,428,430,431,432,433,435,438,442],[510,510,511,511,510,506,500,494,490,489,489,490,495,505,509],[570,570,570,569,569,566,563,562,555,549,546,547,551,554,558,559,561,565,568,570,571,571,572,572,572,574,575],[594,593,593,591,590,590,591,595,600,602,602,602,599,593,592,590,592,597,599,605],[623,623,623,623,623,622,622,622,625,627,631,637,642,644,645,643,642,641,640,640,639,639,639,637,637,635,634,631,625,623],[660,660,660,661,661,660,660,659,659,659,660,661,663,670,673,673,671,670,668,661,660,658,657],[690,689,688,689,692,693,696,698,700,704,709,710],[737,738,738,739,740,740,740,740,739,735,731,726,725,723,722,722,723,725,726,727,731,735,736,739,741,742,743,744,745,747,748],[774,774,773,773,772,772,773,773,773,772,772,770,768,768,769,772,777,778,781,783,784,785,785,781,777,773,772,771,771],[807,806,805,805,804,804,806,809,810,812,814,815,815,815,813,811,810,809,812,814,815,819,820,825,827,828,828,828,825,824,823,822,822,825],[841,842,843,843,844,845,845,845,845,845,844,843,844,846,847,850,850,852,852,851,848,846],[882,881,880,881,882,887,894,895,898,900],[925,925,925,925,925,925,925,925,924,923,922,922,922,923,925,929,933,934,935,936,937,936,936,933,931,927,926,925,923],[959,957,957,956,955,954,953,952,951,951,950,948,948,948,948,949,951,957,959,963],[612,614,622,627,645,660,675,715,726,748,783,795,830,842,870,886,899,904,911,913,915,914,912,909,905,902],[690,691,692,692,693,693,692,689,687,682,677,672,671,670,668,668,668,669,672,673,677,679,680,683,685,689,690,693,694,695,695,696,698,699],[720,718,717,717,719,720,724,726,726,727,727,725,724,721,720,717,718,724,730],[752,751,750,749,753,760,762,766,770,771],[763,763,763,763,762,762,762,762,761,761,761,761,761,761],[805,804,804,803,802,802,802,802,802,801,801,801,799,799,798,799,803,804,806,809,810,812,812,813,812,811,808,805,803,801,799],[829,829,828,828,828,829,830,832,835,837,837,838,837,835,832,831,829,827,829,836,839,844],[1001,1000,1000,1000,1002,1004,1007,1009,1010,1013,1013,1008,1006,998,995,986,980,977,971,969,965]],"y":[[128,128,128,127,127,125,124,123,123,123,124,130,137,144,149,153,152,151],[125,125,125,124,124,123,124,126,129,133,135,139,145,150,149,148],[152,151,150,149,150,152,156,158,165,172,179,181],[125,125,124,124,123,123,123,123,123,123,123],[139,139,138,138,137,135,135,134],[132,132,132,132,132,132,132,132,132,131,130,130,130],[124,122,120,119,118,118,117,117,122,128,135,140,140,140,140,140,139],[168,169,169,169,169,169,170,170,169,169,168,167,167,166,166,167],[201,200,199,198,196,195,193,192,192,193,195,196,202,208,209,212,213,213,212,209,208,206,205,204,204,206,207,208,209,209],[149,149,149,148,148,148,148,149,148,147],[122,117,116,115,114,117,126,129,134,142,144,151,154,154,152,152,150,150,150,150,150,151,152,154,155,156,157,157,158,158,157,156],[179,178,178,178,178,177,177,177,177,178,178],[210,209,208,207,206,202,201,199,198,198,198,201,205,207,208,211,213,214,214,214,213,212,212,211,211,212,214,217,218,217,215],[122,119,117,113,110,109,116,133,150,168,174,191,207,214,214],[129,127,126,125,122,121,122,123,129,137,143,143,141,139,136,134,133,130,129,130,131,132,134,135,138,141,141],[101,99,98,96,95,94,94,94,97,99,103,105,109,115,116,117,117,117,117,116],[124,122,121,120,123,127,131,136,139,140,140,138,134,131,128,127,127,128,133,141,154,157,164,171,173,174,174,173,168,166],[152,150,149,150,154,155,158,162,163,165,166,166,166,161,157,153,151,150,149,148,148,148,148],[132,131,130,130,130,130,130,130,130,130,129,129],[135,133,132,131,128,127,125,124,123,122,124,127,129,134,135,137,139,139,139,139,137,133,132,131,130,130,130,133,135,137,137],[107,105,104,102,101,103,109,111,115,121,123,128,133,134,134,132,130,129,128,128,128,130,132,136,138,139,139,138,137],[123,123,123,122,122,121,121,122,122,123,126,127,129,130,132,135,135,137,135,134,132,130,129,125,123,122,121,122,124,126,128,131,134,136],[142,142,142,141,141,140,141,142,143,145,148,152,153,153,152,149,148,145,142,140,139,138],[122,122,122,122,122,122,122,122,122,122],[108,106,105,104,103,106,112,113,119,126,129,130,132,132,131,129,128,128,128,129,129,131,132,134,135,136,136,137,136],[127,125,124,123,122,121,121,1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    Reduciendo a un denominador común, obtenemos

    x1=-c(a2+b2)-ba2y0+ab2x0+b2caa2+b2

    Simplificando, tenemos

    x1=-a2c-b2c-a2by0+ab2x0+b2ca(a2+b2)

    Si simplificamos aún más eliminando los términos semejantes, obtenemos

    x1=b2x0-aby0-aca2+b2

    Ahora hemos obtenido las coordenadas del punto Q en función de las constantes que conocemos,

    Qb2x0-aby0-aca2+b2,a2y0-abx0-bca2+b2

    Ahora podemos calcular la distancia entre A y Q utilizando la distancia, que no es otra cosa que la distancia del punto a la recta , como hemos comentado antes. Denotémosla por d y apliquemos la fórmula de la distancia,

    d2=x1-x02+y1-y02

    Sustituyendo x1,y1 obtenemos

    d2=b2x0-aby0-ac-a2x0-b2x0a2+b2+a2y0-aby0-bc-a2y0-b2y0a2+b2

    Simplificando aún más, obtenemos

    d2=a2ax0+by0+c2+b2ax0+by0+c2(a2+b2)2

    d2=a2+b2ax0+by0+c2a2+b22

    d2=ax0+by0+c2a2+b2

    Tomando la raíz cuadrada de ambos lados, obtenemos

    d=±ax0+by0+ca2+b2

    Como d es distancia, no puede ser negativa, así que rechazamos la raíz negativa, lo que nos da,

    d=ax0+by0+ca2+b2

    Pero sigue existiendo la posibilidad cuando el numerador ax0+by0+c sea negativo. Para evitar que sea negativo, hay que tomar su módulo,

    d=ax0+by0+ca2+b2

    No nos encontramos con ese problema, ya que el denominador es una suma de cuadrados de números distintos de cero, por lo que siempre será positivo.

    Para escribir la misma expresión de una forma más cómoda (y fácil de recordar), definamos la ecuación de la recta como fx,y=ax+by+c para obtener fx0,y0=ax0+by0+cquedando

    d=|fx0,y0|a2+b2

    Apliquemos ahora esta fórmula mediante un par de ejemplos.

    Calcular la distancia de un punto a una recta

    Calcular la distancia de una recta a un punto es un proceso relativamente sencillo. Se dará la ecuación de una recta y el punto hasta el que se debe calcular la distancia.

    Halla la distancia entre la recta 4x+3y-2=0 y el punto 2,4.

    Solución

    Comparando la ecuación dada con la forma general, obtenemos a=4, b=3 y c=-2, y x0=2, y0=4.

    Recordando la fórmula de la distancia y sustituyendo todos los valores correspondientes obtenemos

    d=ax0+by0+ca2+b2=42+34-242+32=185

    Por tanto, la distancia entre la recta 4x+3y-2=0 y 2,4 es 185 unidades.

    Halla la distancia entre la recta 5x-2y=0{"x":[[208,204,202,199,195,190,187,178,175,166,164,156,154,152,149,147,146,144,143,142,141,140,139,139,138,137,135,134,132,131,129,128,126,125,123,123,123,123,123,123,124,125,127,132,135,140,142,151,154,164,168,178,181,185,190,192,194,198,199,202,202,202,200,199,197,193,190,185,179,173,170,163],[260,260,261,263,266,271,274,280,284,287,295,298,304,307,315,317,317,315,313,304,301,297,291,287,283,278,275,273],[346,348,348,349,349,348,345,341,339,332,329,321,319,316,315,315,317,318,321,323,325,330,337,345,353,357],[414,414,415,418,422,427,430,442,445,452,458,466,468,469,470],[508,508,508,508,511,514,519,522,525,528,534,539,543,545,548,549,550,550,548,545,543,535,532,524,519,515,513,512,510,510,510,511,513,514,519,522,528,536,543,552,560,563,573,577],[618,619,620,621,622,622,623,623,623,624,624,625,625,626,628,629,630,634,637,640,644,646,649,654,657,660,664,666,671,675,679,680,683,684,684,683,682,680,678,676,673,671,669,668,667,666,666,665,665,664,663,661,660,656,654,644,640,638,633,632],[757,758,760,762,766,771,774,785,788,794,796,799,804],[732,733,738,741,748,760,764,777,786,794,798,802,805,812],[889,890,890,890,888,887,884,883,880,879,877,875,875,873,873,873,874,877,879,886,889,897,903,907,914,917,920,927,931,933,939,942,944,949,951,954,957,958,958,956,955,949,942,936,932,927,923,915,907,900,897,895,891,889,888,886]],"y":[[227,229,229,230,231,233,234,238,239,241,242,243,243,243,243,242,242,242,242,242,242,243,245,247,249,254,263,271,278,282,293,297,307,310,318,320,322,325,326,327,327,327,325,320,316,312,310,304,303,298,298,298,298,299,302,304,307,312,315,325,332,336,344,348,352,360,363,370,376,380,382,384],[296,294,290,286,283,280,279,277,276,276,276,276,278,280,289,298,301,312,316,329,332,336,343,346,349,353,355,356],[278,277,275,275,274,274,275,278,279,285,288,297,301,312,318,324,330,332,336,338,340,342,343,343,342,341],[293,294,294,294,294,294,293,292,292,291,290,289,289,289,289],[267,266,264,263,260,258,257,256,256,256,257,259,262,264,268,273,276,279,286,293,296,307,311,320,326,331,333,335,338,340,341,342,342,342,342,342,341,339,337,334,332,331,328,327],[269,269,269,269,270,272,277,284,288,303,308,321,324,328,333,336,338,342,344,345,345,345,345,342,340,338,333,331,325,319,313,310,303,302,298,298,300,306,309,316,324,334,345,352,358,371,377,391,401,412,422,429,432,440,442,445,442,441,436,433],[302,302,302,303,303,303,303,303,303,303,303,303,304],[351,351,352,351,350,346,345,342,341,340,340,340,340,339],[256,256,254,253,256,257,266,270,280,287,293,309,315,329,343,356,361,376,381,390,393,398,399,399,397,395,393,388,384,380,370,365,359,347,342,329,317,306,300,284,279,267,260,256,255,254,254,255,258,262,265,267,272,274,277,281]],"t":[[0,10,12,14,25,35,39,51,55,68,72,85,89,89,101,105,118,122,123,130,139,151,156,156,168,172,185,189,201,207,218,223,235,239,251,256,256,264,272,284,289,301,306,318,322,334,340,351,356,368,373,385,389,390,402,406,406,417,422,431,439,452,456,456,468,472,473,485,489,501,507,514],[846,852,857,858,864,872,884,889,889,901,906,906,918,923,940,948,956,967,973,984,989,989,1001,1006,1006,1014,1023,1023],[1188,1195,1199,1201,1206,1218,1223,1235,1240,1251,1256,1268,1273,1284,1289,1301,1306,1318,1323,1323,1335,1339,1351,1356,1368,1370],[1625,1639,1656,1668,1674,1684,1690,1701,1707,1718,1723,1735,1739,1740,1748],[2016,2025,2035,2041,2051,2056,2065,2073,2073,2085,2090,2102,2106,2107,2118,2123,2131,2132,2140,2152,2156,2168,2173,2185,2190,2202,2206,2207,2215,2223,2235,2240,2240,2252,2256,2257,2269,2273,2285,2290,2302,2306,2318,2324],[2536,2553,2558,2559,2565,2567,2574,2585,2590,2602,2607,2618,2623,2626,2635,2640,2640,2648,2657,2668,2673,2673,2686,2690,2690,2702,2706,2707,2719,2724,2735,2740,2752,2757,2769,2785,2790,2802,2807,2819,2823,2835,2840,2840,2852,2859,2860,2869,2873,2886,2890,2902,2907,2919,2923,2941,2949,2957,2969,2971],[3355,3369,3372,3374,3382,3390,3402,3407,3419,3423,3424,3436,3441],[3610,3625,3629,3634,3640,3652,3657,3669,3674,3686,3690,3691,3702,3707],[3963,3968,3975,3982,4007,4018,4024,4036,4040,4041,4053,4057,4057,4070,4074,4086,4091,4103,4107,4119,4124,4136,4140,4153,4157,4157,4170,4174,4174,4186,4190,4191,4203,4207,4208,4217,4224,4236,4241,4253,4257,4269,4274,4282,4291,4291,4303,4307,4316,4324,4325,4336,4340,4341,4349,4350]],"version":"2.0.0"} y el punto 3,0.

    Solución

    Comparando la ecuación dada con la forma general, obtenemos a=5, b=-2 y c=0, y x1,y1=3,0.

    Recordando la fórmula de la distancia y sustituyendo todos los valores correspondientes obtenemos,

    d=ax0+by0+ca2+b2=5(3)+0+052+(-2)2=1529

    Por tanto, la distancia entre la recta 5x-2y=0 y (3,0) es de 1529 unidades.

    Ejemplo de distancia de un punto a una recta

    Halla la distancia entre las dos rectas cuyas ecuaciones vienen dadas por y=2x-5 y y=2x+3.

    Solución

    Observa que las dos rectas tienen la misma pendiente, 2, lo que implica que son paralelas.

    Para hallar la distancia entre las dos rectas, podemos tomar un punto de una de ellas y utilizar la fórmula "distancia de un punto a una recta" para obtener la distancia.

    Busquemos el punto de la primera recta y=2x-5 sustituyendo x=0 (como es fácil de calcular, cualquier otro valor de x habría sido igualmente válido),

    y=2(0)-5=-5

    Así, un punto de la recta es (0,-5). Ahora podemos utilizar la fórmula que hemos obtenido antes para hallar la distancia entre este punto y la recta y=2x+3.

    d=ax1+by1+ca2+b2

    Pero primero, reescribimos y=2x+3 en la forma general para obtener, -2x+y-3=0, y por tanto tenemos a=-2, b=1 y c=-3, que sustituimos en la ecuación anterior para obtener d:

    d=0-5-35=85

    Por tanto, la distancia entre el punto (0,-5) y la recta y=2x+3 es 85 unidades.

    Pero recuerda que este punto se encuentra sobre la recta y=2x-5 y estas rectas son paralelas, por lo que la distancia entre las dos rectas también es 85 como se ha descrito anteriormente.

    Distancia de un punto a una recta - Puntos clave

      • La distancia entre un punto y una recta puede medirse de muchas formas distintas, pero la distancia más corta entre ellos es la que cuenta.
      • La distancia entre un punto y una recta se mide por la distancia más corta entre ellos, también es lo mismo que la distancia perpendicular entre ellos.
      • Sea una recta dada por ax+by+c=0entonces la distancia entre esta recta y el punto x1,y1 viene dada por la fórmula d=ax1+by1+ca2+b2.
    Preguntas frecuentes sobre Distancia de un Punto a una Línea
    ¿Cómo se calcula la distancia de un punto a una línea?
    La distancia de un punto a una línea se calcula utilizando la fórmula: distancia = |Ax1 + By1 + C| / √(A² + B²).
    ¿Qué representa la distancia de un punto a una línea?
    La distancia representa la longitud del segmento perpendicular desde el punto hasta la línea.
    ¿Por qué es importante saber la distancia de un punto a una línea?
    Es importante en geometría y aplicaciones prácticas como diseño y navegación para determinar posiciones y relaciones espaciales.
    ¿Qué variables se necesitan para calcular la distancia?
    Se necesitan las coordenadas del punto (x1, y1) y los coeficientes A, B, C de la ecuación de la línea Ax + By + C = 0.

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