Let tangent drawn to parabola y ^{2}=24 x at point (α, β) is perpendicular to line 2 x +2 y=5 . Th

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10-2. Parabola, Ellipse, Hyperbola

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Let the tangent drawn to the parabola $y ^{2}=24 x$ at the point $(\alpha, \beta)$ is perpendicular to the line $2 x$ $+2 y=5$. Then the normal to the hyperbola $\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$ at the point $(\alpha+4, \beta+4)$ does $NOT$ pass through the point.

A

$(25,10)$

B

$(20,12)$

C

$(30,8)$

D

$(15,13)$

(JEE MAIN-2022)

Solution

Tangent at $(\alpha, \beta)$ has slope 1

$\beta^{2}=24 \alpha$

Equation of tangent $y \beta=12(x+\alpha), \frac{12}{\beta}=1$

$\Rightarrow \alpha=6, \beta=12$

$\therefore(\alpha+4, \beta+4)=(10,16)$

Normal at $(10,16)$ to $\frac{x^{2}}{36}-\frac{y^{2}}{144}=1$ is

$2 x+5 y=100$

Standard 11

Mathematics

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