Question

Points P and Q are two points in the coordinate plane. The x-coordinate of the point equidistant from the points P and Q and lying on the line y = -1 is ___.

Answer 1

The required point is equidistant from the points P and Q, and also lies on the line y = -1.

Consider a point A(x_a, -1), such that it lies on the line y = -1.

Distance AP = Distance AQ

$\Rightarrow \sqrt{(x_a-1{)}^2+(-1-3{)}^2})=\sqrt{(x_a+5{)}^2+(-1+5{)}^2}\Rightarrow {(x_a-1)}^2+(-4{)}^2={(x_a+5)}^2+(4{)}^2\Rightarrow -2(x_a)+1=10x_a+25\Rightarrow -12(x_a)=24\Rightarrow x_a=-2$
The coordinates of the point A is (-2, -1).