Question

Tangents are drawn from the origin to the curve \(y = \sin x\). Then, the point of contact lie on the curve:

• A. \(y^2 = \frac{x^2}{1 - x^2}\)
• B. \(y^2 = \frac{x^2}{1 + y}\)
• C. \(x^2 = \frac{1 + y^2}{y^2}\)
• D. \(y^2 = \frac{x^2}{1 + x^2}\)

Hint:

The Gaussian beam profile ensures minimal divergence and highest spatial coherence.

Answer 1

The Correct Option is D

Slope of tangent from origin is \(y/x\). For curve \(y = \sin x\), equation of tangent: \[ y = \sin a + \cos a (x - a) \Rightarrow 0 = \sin a - a \cos a \Rightarrow \tan a = a \Rightarrow y = \sin a, x = a \Rightarrow y^2 = \frac{x^2}{1 + x^2} \]