Let $A(2 \sec \theta, 3 \tan \theta)$ and $B(2 \sec \phi, 3 \tan \phi)$ where $\theta+\phi=\frac{\pi}{2}$ be two points on the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$. If $(\alpha, \beta)$ is the point of intersection of normals to the hyperbola at $A$ and $B$, then $\beta$ is equal to

Question

Let $A(2 \sec \theta, 3 \tan \theta)$ and $B(2 \sec \phi, 3 \tan \phi)$ where $\theta+\phi=\frac{\pi}{2}$ be two points on the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$. If $(\alpha, \beta)$ is the point of intersection of normals to the hyperbola at $A$ and $B$, then $\beta$ is equal to, $\frac{12}{3}$, $\frac{13}{3}$, $-\frac{12}{3}$, $-\frac{13}{3}$

MARKS App · Accepted Answer

Hint : Equation of Normal at $A$ $\frac{4 x}{2 \sec \theta}+\frac{9 y}{3 \tan \theta}=13$ at $B$ $\begin{aligned} & \frac{4 x}{2 \operatorname{cosec} \theta}+\frac{9 y}{3 \cot \theta}=13 \ & \Rightarrow(2 \cos \theta) x+(3 \cot \theta) y=13 \quad ----(1)\ & \&~(2 \sin \theta) x+(3 \tan \theta) y=13 \quad ---(2)\ & \Rightarrow y=-\frac{13}{3} \end{aligned}$

Search any question & find its solution

Looking for more such questions to practice?