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

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

$\Rightarrow 2x+2y{y}_1-2f{y}_1=0$

$\Rightarrow x=y_1\left(f-y\right)$

$\Rightarrow x=y_1\left(\frac{x^2+y^2-a^2}{2y}-y\right)$

$\Rightarrow 2xy=y_1\left(x^2-y^2-a^2\right)$

$\Rightarrow y_1\left(y^2-x^2+a^2\right)+2xy=0$