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 a2 + b2 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 $\frac{x^2}{a^2} + \frac{y^2}{a^2} = 1$
a2 + b2 = 25/9