Question

The differential equation of the family of circles passing through the points $(0,2)$ and $(0,-2)$ is

• A. $2xy\frac{dy}{dx}+\left(x^2-y^2+4\right)=0$
• B. $2xy\frac{dy}{dx}+\left(x^2+y^2-4\right)=0$
• C. $2xy\frac{dy}{dx}+\left(y^2-x^2+4\right)=0$
• D. $2xy\frac{dy}{dx}-\left(x^2-y^2+4\right)=0$

Answer 1

Answer: (A)

Sol. Equation of circle passing through $(0,-2)$ and $(0,2)$ is
$x^2+\left(y^2-4\right)+\lambda x=0,(\lambda \in R)$
Divided by $x$ we get
$\frac{x^2+\left(y^2-4\right)}{x}+\lambda =0$
Differentiating with respect to $x$
$\frac{x\left[2x+2y\cdot \frac{dy}{dx}\right]-\left[x^2+y^2-4\right]\cdot 1}{x^2}=0$
$\Rightarrow 2xy\cdot \frac{dy}{dx}+\left(x^2-y^2+4\right)=0$