$Q$ is a fixed point and $S$ is a fixed circle . A variable chord through $Q$ meets the circle at A and B. Find locus of P on this chord such that $QA,QP,QB$ are in Arithmetic Progression.
Could someone give me slight hint as how to approach this problem?
On
If point $Q$ is outside circle $S$, then the answer is easy, because $P$ is the midpoint of $AB$.
A more interesting case arises if $Q$ is inside $S$. Without loss of generality, one may assume $Q$ to be the origin of a cartesian coordinates system, and the center $C=(x_0,0)$ of $S$ to lie on the $x$-axis. If $R$ is the radius of $S$, then the polar equation of the locus can be written as $$ r(\theta)=\sqrt{R^2-x_0^2\sin^2\theta}. $$ In rectangular coordinates this can be written as an implicit equation of fourth degree: $$ (x^2+y^2)^2+x_0^2y^2=R^2(x^2+y^2). $$
$QA\cdot QB$ is constant, $QP=(QA+QB)/2$.