Question

The differential equation representing a family of circles touching the $y$-axis at the origin is

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

Answer 1

Answer: (C)

$x^2-y^2+2xy\frac{dy}{dx}=0$

$(x-a{)}^2+y^2=a^2$

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

$x^2-2ax+y^2=0$...(1)

Differentiating w.r.t x

$2x-2a+2y\frac{dy}{dx}=0$

$\Rightarrow a=x+y\frac{dy}{dx}$

Substituting the value of a in eqn (1), we get,

$x^2-2x(x+y\frac{dy}{dx})+y^2=0$

$x^2-2x^2-2xy\frac{dy}{dx}+y^2=0$

$y^2-x^2-2xy\frac{dy}{dx}=0$

$\Rightarrow x^2-y^2+2xy\frac{dy}{dx}=0$