Let a line L: 2 x+y=k, k,>,0 be a tangent to hyperbola x^{2}-y^{2}=3 . If L is also a tangent to

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10-2. Parabola, Ellipse, Hyperbola

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Let a line $L: 2 x+y=k, k\,>\,0$ be a tangent to the hyperbola $x^{2}-y^{2}=3 .$ If $L$ is also a tangent to the parabola $y^{2}=\alpha x$, then $\alpha$ is equal to :

A

$24$

B

$-12$

C

$-24$

D

$12$

(JEE MAIN-2021)

Solution

Tangent to hyperbola of

Slope $\mathrm{m}=-2$ (given)

$y=-2 x \pm \sqrt{3(3)}$

$\left(y=m x \pm \sqrt{a^{2} m^{2}-b^{2}}\right)$

$\Rightarrow y+2 x=\pm 3 \Rightarrow 2 x+y=3(k\,>\,0)$

For parabola $y^{2}=a x$

$y=m x+\frac{\alpha}{4 m}$

$\Rightarrow y=-2 x+\frac{\alpha}{-8}$

$\Rightarrow \frac{\alpha}{-8}=3$

$\Rightarrow=-24$

Standard 11

Mathematics

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