Question

If \(\sin^{-1}\left(\frac{x^2 - y^2}{x^2 + y^2}\right) = \log a\), then \(\frac{d^2y}{dx^2}\) equals

• A. \(\frac{y}{x^2}\)
• B. \(\frac{x}{y^2}\)
• C. \(-\frac{y}{x^2}\)
• D. 0

Hint:

Use substitution \(y = x\tan\theta\) to simplify.

Answer 1

The Correct Option is D

Step 1: Formula / Definition}
\[ \frac{x^2 - y^2}{x^2 + y^2} = \sin(\log a) = c \text{ (constant)} \]
Step 2: Calculation / Simplification}
\(\frac{1 - (y/x)^2}{1 + (y/x)^2} = c\)
Let \(y = x\tan\theta\): \(\frac{1-\tan^2\theta}{1+\tan^2\theta} = \cos 2\theta = c\)
\(\theta = \frac{1}{2}\cos^{-1}c\) (constant)
\(y = x\tan\theta \Rightarrow \frac{dy}{dx} = \tan\theta = \frac{y}{x}\)
\(\frac{d^2y}{dx^2} = \frac{x(dy/dx) - y}{x^2} = \frac{x(y/x) - y}{x^2} = 0\)
Step 3: Final Answer
\[ 0 \]