Let mathrm{A},(sec θ, 2 tan θ) and mathrm{B},(sec phi, 2 tan phi) , where θ+phi=π / 2 , be two p

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10-2. Parabola, Ellipse, Hyperbola

hard

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

A

$6$

B

$12$

C

$24$

D

None of these

(JEE MAIN-2021)

Solution

Since, point $A(\sec \theta, 2 \tan \theta)$

lies on the hyperbola

$2 x^{2}-y^{2}=2$

Therefore, $2 \sec ^{2} \theta-4 \tan ^{2} \theta=2$

$\Rightarrow 2+2 \tan ^{2} \theta-4 \tan ^{2} \theta=2$

$\Rightarrow \tan \theta=0 \Rightarrow \theta=0$

Similarly, for point $\mathrm{B}$, we will get $\phi=0$.

but according to question $\theta+\phi=\frac{\pi}{2}$

which is not possible.

Standard 11

Mathematics

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