Question

The differential equation of all circles which pass through the origin and whose centers lie on the y-axis is

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

Answer 1

The correct option is A $(x^2-y^2)\frac{dy}{dx}-2xy=0$

If $(0,k)$ be the centre on y-axis then its radius will be k as it passes through origin. Hence its equation is
$x^2+(y-k{)}^2=k^2$
Or $x^2+y^2=2ky$ (1)
$\therefore 2x+2y\frac{dy}{dx}=2k\frac{dy}{dx}$
$=\frac{x^2+y^2}{y}\frac{dy}{dx}$ [by(1)]
$\therefore 2xy=x^2+y^2-2y^2)\frac{dy}{dx}$
or $(x^2-y^2)\frac{dy}{dx}-2xy=0$