Question

Se pertence ao eixo das abscissas e é equidistante dos pontos A(1,4) B(-6,3) determine as coordenadas de P

Answer 1

Se P está equidistante de A e B, então a distancia AP e distancia BP são iguais.

Ainda, se B pertence ao eixo das abscissas (eixo x), sua coordenada "y" vale zero.

Equacionando as distancias:

$Distancia_{AP}~=~Distancia_{BP}\\\\\\\sqrt{(x_A-x_P)^2+(y_A-y_P)^2}~=~\sqrt{(x_B-x_P)^2+(y_B-y_P)^2}\\\\\\\sqrt{(1-x_P)^2+(4-0)^2}~=~\sqrt{(-6-x_P)^2+(3-0)^2}\\\\\\\sqrt{(x_P^2-2x_P+1)+(4)^2}~=~\sqrt{(x_P^2+12x_P+36)+(3)^2}\\\\\\(x_P^2-2x_P+1)+(4)^2~=~(x_P^2+12x_P+36)+(3)^2\\\\\\x_P^2-2x_P+1+16~=~x_P^2+12x_P+36+9\\\\\\12x_P+2x_P~=~1+16-36-9\\\\\\14x_P~=~-28\\\\\\x_P~=~\frac{-28}{14}\\\\\\\boxed{x_P~=~-2}$

Resposta:   P = (-2 , 0)