Question

If a point $A(0,2)$ is equidistant from the points $B(3,p)$ and $C(p,5)$ then find the value of $p.$

Answer 1

Distance between two points=$\sqrt{(x_2-x_1{)}^2(y_2-y_1{)}^2}$

$\therefore AB=\sqrt{(0-3{)}^2+(2-p{)}^2}$

$\Rightarrow \sqrt{9+4-4p+p^2}$

$\Rightarrow \sqrt{p^2-4p+13}$

$\therefore AC=\sqrt{(p-0{)}^2+(5-2{)}^2}$

$\Rightarrow \sqrt{p^2+9}$

As given $AB=AC$

$\Rightarrow \sqrt{p^2-4p+13}=\sqrt{p^2+9}$

Squaring both sides

$\Rightarrow p^2-4p+13=p^2+9$

$\Rightarrow -4p=9-13$

$\Rightarrow -4p=-4$

$\Rightarrow p=\frac{-4}{-4}=1$