Hyperbola
Intersection of Normals
IIT-JEE 1999
Grade 11
Question:
Let $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$, with $\theta + \phi = \pi/2$, be points on $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If $(h, k)$ is the intersection of normals at $P$ and $Q$, then $k$ equals:
(1) $\frac{a^2 + b^2}{a}$
(2) $-\frac{a^2 + b^2}{a}$
(3) $\frac{a^2 + b^2}{b}$
(4) $-\frac{a^2 + b^2}{b}$
Step-by-Step Solution
Key Concept: Normal at $P(a \sec \theta, b \tan \theta)$: $ax \cos \theta + by \cot \theta = a^2 + b^2$. Since $\phi = \pi/2 - \theta$: $\sec \phi = \csc \theta$, $\tan \phi = \cot \theta$. By symmetry the normals intersect at $k = -(a^2 + b^2)/b$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (4)