The Differential Equation representing the family of tangents to the circle $x^2+y^2 =r^2$, is :

$\frac{d^2y}{dx^2}=0$

The correct answer is Option (2) → S$\frac{d^2y}{dx^2}=0$

$x^2+y^2 =r^2$

so differentiating wrt x

we get $2x+2y\frac{dy}{dx}=0$

$\frac{dy}{dx}=-frac{x}{y}$

let point of tangent be (a, b)

eq:- $y-b=-\frac{a}{b}(x-a)$

$by-b^2=-ax+a^2$

$ax+by=a^2+b^2$

differentiating wrt x 

$a+b\frac{dy}{dx}=0$

so $\frac{dy}{dx}=-\frac{a}{b}$

$⇒\frac{d^2y}{dx^2}=0$