The differential equation of family of circles having centre on line $y = 10$ and touching x-axis is

The general equation for a circle can be written as $x^2 + y^2 + 2gx + 2fy + c = 0$

The center lies on $y = 10$, and so $f = -10.$

Also, distance of the center from x axis has to be equal to the radius.

$\Rightarrow f^2 = g^2 + f^2 - c$

$\Rightarrow g^2 = c$

The equation thus becomes $x^2 + y^2 + 2gx - 20y + g^2 = 0$

Differentiating w.r.t. x, we get $2x + 2y \dfrac{dy}{dx} + 2g - 20 \dfrac{dy}{dx} = 0$

$\Rightarrow g = (10 - y) \dfrac{dy}{dx} - x$

Resubstituting this value in the above equation, we get $x^2 + y^2 - 20y + 2x\left[(10 - y) \dfrac{dy}{dx} - x\right] + \left[(10 - y) \dfrac{dy}{dx} - x\right]^2$

$= x^2 + y^2 - 20y + 2x(10 - y) \dfrac{dy}{dx} - 2x^2 + (10 - y)^2 \left(\dfrac{dy}{dx}\right)^2 + x^2 - 2x(10 - y) \dfrac{dy}{dx}$

$= y^2 - 20y + (10 - y)^2\left(\dfrac{dy}{dx}\right)^2$