Question

The differential equation of the family of circles passing through the fixed point (a, 0) and (-a, 0) is

• A. $y_1(y^2-x^2)+2xy+a^2=0$
• B. $y_1{y}^2+xy+a^2{x}^2=0$
• C. $y_1(y^2-x^2+a^2)+2xy=0$
• D. $y_1(y^2+x^2)-2xy+a^2=0$

Answer 1

Answer: (C)

$y_1(y^2-x^2+a^2)+2xy=0$

Let the equation of circle passing through given points is
$x^2+y^2-2fy=a^2$
$\Rightarrow 2x+2{yy}_1-2{fy}_1=0$
$\Rightarrow x=y_1(f-y)$
$\Rightarrow x=y_1(\frac{x^2+y^2-a^2}{2y}-y)$
$\Rightarrow 2xy=y_1(x^2-y^2-a^2)$
$\Rightarrow y_1(y^2-x^2+a^2)+2xy=0$