Q: A tangent of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is normal to the hyperbola $\frac{x^2}{4} – \frac{y^2}{1} = 1$ and it has equal intercepts with positive x and y axes, then the value of a^{2} + b^{2} is

(A) 5

(B) 25

(C) 16

(D) 25/9

Sol. The equation of normal to the hyperbola $\frac{x^2}{4} – \frac{y^2}{1} = 1$ at (2 sec θ , tan θ) is 2x cos θ + y cot θ = 5

Slope of normal = – 2 sin θ = – 1

⇒ θ = π/6

y-intercept of normal = 5/cotθ = 5/√3

Since it touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{a^2} = 1$

a^{2} + b^{2} = 25/9