Question

If \( x = a t^2, \; y = 2at \), then \( \frac{d^2 y}{dx^2} =\) _____

• A. \( \frac{a}{xy} \)
• B. \( \frac{ax}{y} \)
• C. \( -\frac{a}{xy} \)
• D. \( -\frac{ax}{y} \)

Hint:

Always convert parametric derivatives into \(x,y\) form at the end.

Answer 1

The Correct Option is C

Concept: For parametric equations: \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Step 1: Differentiate. \[ \frac{dx}{dt} = 2at, \quad \frac{dy}{dt} = 2a \] \[ \frac{dy}{dx} = \frac{2a}{2at} = \frac{1}{t} \]
Step 2: \[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{1}{t}\right) = \frac{d}{dt}\left(\frac{1}{t}\right) \cdot \frac{dt}{dx} \] \[ = -\frac{1}{t^2} \cdot \frac{1}{2at} = -\frac{1}{2at^3} \]
Step 3: Express in \(x,y\): \[ x = at^2, \quad y = 2at \] \[ \Rightarrow t = \frac{y}{2a} \] Substituting gives: \[ \frac{d^2y}{dx^2} = -\frac{a}{xy} \]