Question

Detrmine os valores de M para os quais a distância entre A(-1,3) e B(2,-m) é 9?

Answer 1

A distância entre dois ponto é dada por:

$D_{AB}=\sqrt{\left(x-x_{0}\right)^2+\left(y-y_{0}\right)^2}\\ \\ 9=\sqrt{\left(2-\left(-1\right)\right)^2+\left(-m-3\right)^2}\\ \\ 9=\sqrt{\left(2+1\right)^2+\left(-m-3\right)^2}\\ \\ 9=\sqrt{9+\left(-m-3\right)^2}\\ \\ 9^{2}=\left(\sqrt{9+\left(-m-3\right)^2}\right)^{2}\\ \\ 81=9+\left(-m-3\right)^2\\ \\ \left(-m-3\right)^2=81-9\\ \\ \left(-m-3\right)^2=72\\ \\ \sqrt{\left(-m-3\right)^2}=\pm\sqrt{72}\\ \\ -m-3=\pm\sqrt{3^{2}\cdot 2^{2}\cdot 2}\\ \\ -m=3\pm 6\sqrt{2}$

$m=-3\pm 6\sqrt{2}\\ \\ m_{1}=-3+6\sqrt{2}\ \ ou\ \ 3\left(-1+2\sqrt{2}\right)\\ \\ m_{2}=-3-6\sqrt{2}\ \ ou\ \ 3\left(-1-2\sqrt{2}\right)$