Question

The differential equation of family of circles whose centre lies on $ x $ -axis, is

• A. $ \frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{2} + 1 = 0 $
• B. $ y\frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{2} - 1 = 0 $
• C. $ y\frac{d^{2}y}{dx^{2}}-\left(\frac{dy}{dx}\right)^{2} - 1 = 0 $
• D. $ y \frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{2} + 1 = 0 $

Answer 1

Answer: (D)

The equation of family of circle having centre at
$x$ -axis is $x^{2}+y^{2}-2 a x=0$
On differentiating, we get
$2 x+2 y \frac{d y}{d x}-2 a=0$
Again, differentiating, we get
$ 2+2\left[y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}\right]=0 $
$\Rightarrow y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+1=0 $