Question

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. $y_{1}\left(y^{2}+x^{2}\right)-2xy +a^{2}= 0$

Answer 1

Answer: (C)

Let the equation of circle passing through given points is

$x^2 +y^2 -2fy = a^2$

$\Rightarrow 2x+2yy_{1}-2 fy_{1}=0 $

$\Rightarrow x = y_{1}\left(f-y\right) $

$\Rightarrow x= y_{1}\left(\frac{x^{2}+y^{2}-a^{2}}{2y} -y\right) $

$ \Rightarrow 2xy = y_{1}\left(x^{2}-y^{2}-a^{2}\right) $

$\Rightarrow y_{1}\left(y^{2}-x^{2}+a^{2}\right)+2xy = 0 $