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  4. Let the normal at the point P on the parabola y 2= 6 x pass through the point (5,-8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :

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Q. Let the normal at the point $P$ on the parabola $y ^{2}=$ $6 x$ pass through the point $(5,-8)$. If the tangent at $P$ to the parabola intersects its directrix at the point $Q$, then the ordinate of the point $Q$ is :

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Solution:

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Equation of normal : $y =- tx +2 at + at ^{3} \left( a =\frac{3}{2}\right)$
since passing through $(5,-8)$, we get $t =-2$
Co-ordinate of $Q:(6,-6)$
Equation of tangent at $Q : x +2 y +6=0$
Put $x=\frac{-3}{2}$ to get $R \left(\frac{-3}{2}, \frac{-9}{4}\right)$