Question

The differential equation that represents all parabolas each of which has a latus rectum \( 4a \) and whose axes are parallel to the x-axis is:

• A. \( \frac{d^2y}{dx^2} + \frac{dy}{dx} = 0 \)
• B. \( \frac{d^2y}{dx^2} + \frac{dy}{dx} = 3 \)
• C. \( \frac{d^2y}{dx^2} + \frac{dy}{dx} = 1 \)
• D. \( \frac{d^2y}{dx^2} = 0 \)

Hint:

For parabolas with horizontal axes, the second derivative of the equation yields a simple linear differential equation.

Answer 1

The Correct Option is D

Step 1: Parabola equation. The general form of a parabola with a horizontal axis is \( y^2 = 4ax \), where \( a \) is the latus rectum. The second derivative of this equation gives the differential equation that describes all such parabolas.
Step 2: Conclusion. Thus, the differential equation is \( \frac{d^2y}{dx^2} = 0 \).