Q. The locus of appoint which moves so that the ratio of the length of the tangents to the circles $x^{2} + y^{2} + 4x + 3 = 0$ and $x^{2} + y^{2} - 6x + 5 = 0$ is $2:3$, is
A) $5x^{2} + 5y^{2} - 60x + 7 = 0$
B) $5x^{2} + 5y^{2} + 60x - 7 = 0$
C) $5x^{2} + 5y^{2} - 60x - 7 = 0$
D) $5x^{2} + 5y^{2} + 60x + 7 = 0$
Explanation
(d)
Since, $\frac{\sqrt{S_{1}}}{\sqrt{S_{2}}} = \frac{2}{3}$
$\therefore\ \ \ \frac{\sqrt{x_{1}^{2} + y_{1}^{2} + 4x_{1} + 3}}{\sqrt{x_{1}^{2} + y_{1}^{2} - 6x_{1} + 5}} = \frac{2}{3}$
$\Rightarrow \ 9x_{1}^{2} + 9y_{1}^{2} + 36x_{1} + 27 - 4x_{1}^{2} - 4y_{1}^{2} + 24x_{1} - 20 = 0$
$\Rightarrow \ 5x_{1}^{2} + 5y_{1}^{2} + 60x_{1} + 7 = 0$
$\therefore$ Locus of point is
$5x^{2} + 5y^{2} + 60x + 7 = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.