Question

Let P (a secθ, b tanθ) and Q (a secφ b tanφ) where θ+ φ = P/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 at P and Q, then k is equal to

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

Answer 1

Answer: (D)

$- \left( \frac{a^{2} + b^{2}}{b} \right)$

Answer 2

Normals at θ, φ where φ $\frac{\pi}{2}$ - θ pass through (h, k)

∴ ah cosθ + bk cotθ = a² + b²

and ah sinθ + bk tanθ = a² + b²

Eliminating h, we get k = $- \left( \frac{a^{2} + b^{2}}{b} \right)$.