Question

Let \(P(a \sec \theta, b \tan \theta)\) and \(Q(a \sec \phi, b \tan \phi)\) where \(\theta + \phi = \frac{\pi}{2}\) be two points on the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). If \((h,k)\) is the point of intersection of the normals drawn at \(P\) and \(Q\), then find \(k\).

• A. \(\frac{a^2 + b^2}{a}\)
• B. \(-\frac{a^2 + b^2}{b}\)
• C. \(-\frac{a^2 + b^2}{a}\)
• D. \(\frac{a^2 + b^2}{b}\)

Hint:

Apply the normal form and sum condition \(\theta + \phi = \frac{\pi}{2}\) to find intersection coordinate.

Answer 1

The Correct Option is B

Using parametric form of points on hyperbola and normal equations, solve simultaneous equations to find \[ k = -\frac{a^2 + b^2}{b}. \]