Q. The differential equation of the family of circles passing through the fixed points $(a,0)$ and $( - a,0)$ is
A) $y_{1}\left( y^{2} - x^{2} \right) + 2xy + a^{2} = 0$
B) $y_{1}y^{2} + xy + a^{2}x^{2} = 0$
C) $y_{1}\left( y^{2} - x^{2} + a^{2} \right) + 2xy = 0$
D) $y_{1}\left( y^{2} + x^{2} \right) - 2xy + a^{2} = 0$
Explanation
(c)
Let the equation of circle passing through given points is
$x^{2} + y^{2} - 2fy = a^{2}$
$\Rightarrow$ $2x + 2yy_{1} - 2fy_{1} = 0$ …(i)
$\Rightarrow$ $x = y_{1}(f - y)$
$\Rightarrow$ $x = y_{1}\left( \frac{x^{2} + y^{2} - a^{2}}{2y} - y \right)$ [from Eq. (i)]
$\Rightarrow$ $y_{1}\left( y^{2} - x^{2} + a^{2} \right) + 2xy = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.