Question

If A and B are fixed points in the plane such that $\frac{PA}{PB}=k$ (constant) for all P on a given circle, then the value of k cannot be equal to

Answer 1

Using Apollonian Circles Theorem

Given two point A and B and a number r

The locus of point P such that

$\frac{PA}{PB}=r$ is a circle.

But if $r=1$ , in this case it is the perpendicular bisector of AB

So, for question , if P lies on a circle, then K cannot be equal to 1.