Q. If $P$ is a point such that the ratio of the squares of the lengths of the tangents from $P$ to the circles $x^{2} + y^{2} + 2x - 4y - 20 = 0$ and $x^{2} + y^{2} - 4x + 2y - 44 = 0$ is $2:3$, then the locus of $P$ is a circle with centre
A) $(7,\ - 8)$
B) $( - 7,\ 8)$
C) $(7,\ 8)$
D) $( - 7,\ - 8)$
Explanation
(b)
Let $P(x_{1},y_{2})$ be the point outside the circle. From the given condition
$\frac{x_{1}^{2} + y_{1}^{2} + 2x_{1} - 4y_{1}\text{-}20}{x_{1}^{2} + y_{1}^{2} - 4x_{1} + 2y_{1} - 44} = \frac{2}{3}$
$\Rightarrow 3x_{1}^{2} + 3y_{1}^{2} + 6x_{1} - 12y_{1} - 60 = 2x_{1}^{2} + 2y_{1}^{2} - 8x_{1} + 4y_{1} - 88$
$\Rightarrow \ x_{1}^{2} + y_{1}^{2} + 14x_{1} - 16y_{1} + 28 = 0$
Thus, the locus of point is
$x^{2} + y^{2} + 14x - 16y + 28 = 0$
$\therefore$ Coordinates of centre of circle are $( - 7,\ 8)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.