1. Define the general equation of the circle. 2. Find the DE corresponding to the above equation.

Question

Consider the equation of all circles which pass through the origin and whose centres are on the x-axis.

1. Define the general equation of the circle.
2. Find the DE corresponding to the above equation. 

Answer 1

 1. The general equation of the circle, passing through the origin and whose centers lies on x-axis can be taken as (x – h)² + y² = h² where h being an arbitrary constant.

2. Simplifying (x – h)² + y² = h² we get,

x² – 2hx + h² + y² = h² ⇒ x² – 2hx + h² = 0 _____(1)

Differentiating we get,

2x + 2y \(\frac{dy}{dx}\)  – 2h = 0 ⇒ h = x + y \(\frac{dy}{dx}\)

Substituting in (1) we can eliminate h