Question

The differential equation of all circles of radius $ a $ is:

• A. \( \left( 1 + \frac{dy}{dx} \right)^3 = a^2 \left( \frac{d^2 y}{dx^2} \right)^2 \)
• B. \( \left( 1 - \left( \frac{dy}{dx} \right)^2 \right)^3 = a^2 \left( \frac{d^2 y}{dx^2} \right)^2 \)
• C. \( \left( 1 - \frac{dy}{dx} \right)^3 = a^2 \left( \frac{d^2 y}{dx^2} \right)^2 \)
• D. \( \left( 1 + \frac{dy}{dx} \right)^3 = a^2 \left( \frac{d^2 y}{dx^2} \right)^2 \)

Hint:

For circles, implicitly differentiate the equation twice to derive the differential equation that represents the geometric properties of the curve.

Answer 1

The Correct Option is A

We are asked to find the differential equation of all circles with radius \( a \). A circle's equation can be written as: \[ (x - h)^2 + (y - k)^2 = a^2 \] where \( (h, k) \) is the center and \( a \) is the radius.
Step 1:
Implicitly differentiate the equation of the circle twice with respect to \( x \).
Step 2:
First differentiation gives the first derivative \( \frac{dy}{dx} \), and the second differentiation gives \( \frac{d^2 y}{dx^2} \).
Step 3:
After applying the differentiation process, we arrive at the differential equation: \[ \left( 1 + \frac{dy}{dx} \right)^3 = a^2 \left( \frac{d^2 y}{dx^2} \right)^2 \]