Question

Find the locus of a point such that the line segments having end points(2,0) and (-2,0) subtend a right angle at that point.

Answer 1

$\text{Let P(h,k) be the variable point and letA(2,0) and B(-2,0)be the given points,}Then,\angle APB=\pi /2\Rightarrow A{B}^2=P{A}^2+P{B}^2\Rightarrow (2+2{)}^2+0=(2-h{)}^2+(0-k{)}^2+(-2-h{)}^2+(0-k{)}^2\Rightarrow 16=4+h^2-4h+k^2+4+h^2+4h+k^2\Rightarrow 16+2h^2+2h^2+2k^2+8\Rightarrow 2h^2+2k^2+8-16=0\Rightarrow 2h^2+2k^2-8=0\Rightarrow h^2+k^2-4=0Hence,thelocusof(h,k)is{x}^2+y^2=4.$