Question

encontre o ponto A sabendo que ele pertence ao eixo Y e é equidistante de B (7,8) e C (-3 -5)



Determine o ponto B sabendo que ele pertence ao eixo X e é equidistante de A (3,7) e C (-5 -8)

Por favor me ajudem nessa questão!!! ​

Answer 1

a)

Se A pertence ao eixo "y", sua coordenada "x" vale 0, logo:

$\left\begin{array}{ccc}Distancia_{A,B}&=&Distancia_{A,C}\\\\\\\sqrt{\left(x_A-x_B\right)^2+\left(y_A-y_B\right)^2}&=&\sqrt{\left(x_A-x_C\right)^2+\left(y_A-y_C\right)^2}\\\\\\\sqrt{\left(0-7\right)^2+\left(y-8\right)^2}&=&\sqrt{\left(0-(-3)\right)^2+\left(y-(-5)\right)^2}\\\\\\\sqrt{49+\left(y^2-16y+64\right)}&=&\sqrt{9+\left(y^2+10y+25\right)}\\\\\\49+\left(y^2-16y+64\right)&=&9+\left(y^2+10y+25\right)\\\\\\26y&=&79\end{array}\right$

$\,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{y~~~=~~~\dfrac{79}{26}}\\\\\\~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{A~=~\left(0~,~\frac{79}{26}\right)}$

b)

Se B pertence ao eixo "x", sua coordenada "y" vale 0, logo:

$\left\begin{array}{ccc}Distancia_{B,A}&=&Distancia_{B,C}\\\\\\\sqrt{\left(x_B-x_A\right)^2+\left(y_B-y_A\right)^2}&=&\sqrt{\left(x_B-x_C\right)^2+\left(y_B-y_C\right)^2}\\\\\\\sqrt{\left(x-3\right)^2+\left(0-7\right)^2}&=&\sqrt{\left(x-(-5)\right)^2+\left(0-(-8)\right)^2}\\\\\\\sqrt{\left(x^2-6x+9\right)+49}&=&\sqrt{\left(x^2+10x+25\right)+64}\\\\\\\left(x^2-6x+9\right)+49&=&\left(x^2+10x+25\right)+64\\\\\\16x&=&-31\end{array}\right$

$\,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{x~~~=~~\,-\dfrac{31}{16}}\\\\\\~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{B~=~\left(-\frac{31}{16}~,~0\right)}$