Question

Let $A \equiv(-3,0)$ and $B \equiv(3,0)$ be two fixed points and $P$ moves on a plane such that $PA = nPB ( n >0)$.
If $0< n< 1$, then -

• 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

Answer: (A)

For $0< n< 1 $
locus is $\left(1-n^2\right)\left(x^2+y^2\right)+6 x\left(1+n^2\right)+9\left(1-n^2\right)=0$
putting $A(-3,0)$ in the above equation $9\left(1-n^2\right)-18\left(1+n^2\right)+9\left(1-n^2\right)=-36 n^2< 0$
$\therefore$ A lies inside the circle.
Similarly for B $(3,0)$
$9\left(1-n^2\right)+18\left(1+n^2\right)+9\left(1-n^2\right) $
$ =36 >0$
$\therefore$ B lies outside the circle.