Question

Let $A\equiv (-3,0)$ and $B\equiv (3,0)$ be two fixed points and $P$ moves on a plane such that $PA=n\times PB(n>0)$. If $0<n<1$, then the locus of point $P$ is a circle and

• A. $A$ lies inside the circle and $B$ lies outside the circle
• B. $A$ lies outside the circle and $B$ lies inside the circle
• C. both $A$ and $B$ lies on the circle
• D. both $A$ and $B$ lies inside the circle

Answer 1

The correct option is A $A$ lies inside the circle and $B$ lies outside the circle

For $0<n<1$
locus is $(1-n^2)(x^2+y^2)+6x(1+n^2)+9(1-n^2)=0$

putting $A(-3,0)$ in the above equation
$9(1-n^2)-18(1+n^2)+9(1-n^2)=-36n^2<0$

$\therefore$ $A$ lies inside the circle.

Similarly for $B(3,0)$
$9(1-n^2)+18(1+n^2)+9(1-n^2)$ $=36>0$

$\therefore$ $B$ lies outside the circle.