Question

If the normal to the rectangular hyperbola $x^2 - y^2 = 1$ at the point $P(\tfrac{\pi}{4})$ meets the curve again at $Q(\theta)$, then $\sec^2\theta + \tan\theta =$

• A. $43$
• B. $57$
• C. $3$
• D. $1$

Hint:

Use the parametric form for rectangular hyperbolas and geometric interpretation of normals.

Answer 1

The Correct Option is B

Standard parametric form of rectangular hyperbola $x^2 - y^2 = 1$ is $x = \sec\theta, y = \tan\theta$
Given $P = (\sec(\tfrac{\pi}{4}), \tan(\tfrac{\pi}{4})) = (\sqrt{2}, 1)$
Find equation of normal and second point of intersection $Q(\theta)$
From identity and computation we get $\sec^2\theta + \tan\theta = 57$