The differential equation of all circles touching x-axis at origin is

Correct option is D

​Solution:

$$\textbf{Equation of family of circles:} \\x^2 + (y - b)^2 = b^2 \Rightarrow x^2 + y^2 - 2by = 0 \\[6pt]\textbf{Differentiate both sides w.r.t. } x: \\\frac{d}{dx}(x^2 + y^2 - 2by) = 0 \\\Rightarrow 2x + 2y y' - 2b y' = 0 \\[6pt]\text{From original: } x^2 + y^2 = 2by \Rightarrow b = \frac{x^2 + y^2}{2y} \\[6pt]\text{Substitute: } \\2x + 2y y' - \left( \frac{x^2 + y^2}{y} \right) y' = 0 \\\Rightarrow 2x = y' \cdot \left( \frac{x^2 - y^2}{y} \right) \\\Rightarrow (x^2 - y^2) y' = 2xy \\[10pt]\boxed{\text{Answer: (D) } (x^2 - y^2) y' = 2xy}$$​​