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  4. Let the tangent drawn to the parabola y 2=24 x at the point (α, β) is perpendicular to the line 2 x+2 y=5. Then the normal to the hyperbola (x2/α2)-(y2/β2)=1 at the point (α+4, β+4) does NOT pass through the point :

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Q. 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 :

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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
$2 x+5 y=100$