Question

Determine os valores de m para os quais a distância entre A (m-1, 3) e B (2, m) é 6

Answer 1

$d_{AB}= \sqrt{(x_B-x_A)^2+(y_B-y_A)^2}\\\\x_A=m-1\ ......\ y_A=3\\x_B=2\ ......\ y_B=m\\\\\\6= \sqrt{[2-(m-1)]^2+(m-3)^2}\\\\6= \sqrt{(3-m)^2+(m-3)^2}\\\\\\(6)^2=\bigg( \sqrt{(3-m)^2+(m-3)^2}\bigg) ^2\\\\\\36=9-6m+m^2+m^2-6m+9\\36=2m^2-12m+18\\2m^2-12m-18=0\\m^2-6m-9=0\\\\a=1\ ......\ b=-6\ ......\ c=-9\\\\\triangle=b^2-4ac\\\triangle=(-6)^2-4\cdot(1)\cdot(-9)\\\triangle=36+36\\\triangle=72\\\\m= -\dfrac{b \frac{+}{-} \sqrt{\triangle} }{2a}\\\\\\m= \dfrac{6 \frac{+}{-}6 \sqrt{2}}{2}$


$m'= \dfrac{6+6\sqrt{2} }{2}\\\\\\m'=3+3 \sqrt{2}\\\\\\m"= \dfrac{6-6 \sqrt{2} }{2}\\\\\\m"=3-3 \sqrt{2}$