Question

If the tangent at the point $(2\sec \theta ,3\tan \theta )$ to the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$ is parallel to $3x-y+4=0$, then the value of $\theta$, is :

• A. ${45}^o$
• B. ${60}^o$
• C. ${30}^o$
• D. ${75}^o$

Answer 1

Answer: (C)

${30}^o$

Equation of slope at $(2\sec \theta ,3\tan \theta )$ to the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$ is given by,
$\frac{x\sec \theta }{2}-\frac{y\tan \theta }{3}=1..\left(1\right)$
Given line (1) is parallel to $3x-y+4=0$, so slope of both line will be same
$\Rightarrow \frac{3}{2\sin \theta }=3\Rightarrow \sin \theta =\frac{1}{2}\Rightarrow \theta ={30}^o$