Question

The differential equation of the family of circles passing through origin and having centers on x-axis is:

• A. \( 2xy \frac{dy}{dx} + x^2 - y^2 = 0 \)
• B. \( \left( \frac{dy}{dx} \right)^2 + \frac{d^2 y}{dx^2} + 1 = 0 \)
• C. \( xy \frac{dy}{dx} + y^2 - x^2 = 0 \)
• D. \( \frac{dy}{dx} = \frac{x + y}{x - y} \)

Hint:

When working with families of curves (circles, ellipses), differentiate their general equation implicitly and use given geometric properties like passing through a point or axis-aligned centers to eliminate parameters.

Answer 1

The Correct Option is A

Equation of family of circles with centers on x-axis and passing through origin: \[ (x - a)^2 + y^2 = a^2 \Rightarrow x^2 - 2ax + y^2 = 0 \] Differentiate both sides: \[ 2x - 2a + 2y \frac{dy}{dx} = 0 \Rightarrow a = x + y \frac{dy}{dx} \] Substitute \( a \) back: \[ x^2 - 2(x + y \frac{dy}{dx})x + y^2 = 0 \Rightarrow x^2 - 2x^2 - 2xy \frac{dy}{dx} + y^2 = 0 \Rightarrow -x^2 - 2xy \frac{dy}{dx} + y^2 = 0 \Rightarrow 2xy \frac{dy}{dx} + x^2 - y^2 = 0 \]