Q.The differential equation of all circles having their centres on the line $y=5$ and touching the X-axis is... (A) $y^2\left(1+\dfrac{dy}{dx}\right)=25$ (B) $(y-5)^2\left[1+\left(\dfrac{dy}{dx}\right)^2\right]=25$ (C) $(y-5)^2+\left[1+\left(\dfrac{dy}{dx}\right)^2\right]=25$ (D) $(y-5)^2\left[1-\left(\dfrac{dy}{dx}\right)^2\right]=25$

Given a family of curves or relations written with one or more arbitrary constants (for example y = Ae^(3x) + Be^(-3x), or a circle family with a free radius), a differential equation satisfied by every member of that family is formed by differentiating the relation once for each arbitrary constant present, and then eliminating the constants algebraically between the original relation and its derivatives. With one constant, a single differentiation usually already isolates the constant so it can be substituted straight back into the original relation. With two constants, two differentiations are needed, and the elimination typically proceeds by solving one of the derived equations for one constant (or a combination of both), substituting into the other, and simplifying until no constant remains — leaving a relation purely in x, y, and the derivatives up to the order that matches the number of constants eliminated. This is the reverse process to so …