Q. If $P$ is a point such that the ratio of the square 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)
Since, $\frac{S_{1}}{S_{2}} = \frac{x_{1}^{2} + y_{1}^{2} + 2x_{1} - 4y_{1} - 20}{x_{1}^{2} + y_{1}^{2} - 4x_{1} + 2y_{1} - 44} = \frac{2}{3}$
$\Rightarrow \ \ \ x_{1}^{2} + y_{1}^{2} + 14x_{1} - 16y_{1} + 28 = 0$
$\therefore$ Locus of point $P$ is
$x^{2} + y^{2} + 14x - 16y + 28 = 0$
Centre of the circle is $( - 7,\ 8)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.