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Equation of Normal at Given Point

If normal at ...

Question

If normal at p to parabola $y^{2} = 4 a x$ meets parabola again at Q such that PQ subtends a right angle at the vertex of $y^{2} = 4 a x$ then p will be

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Solution

The correct option is C

It is useful to remember that fact that,If normal at

$(a t_{1}^{2}, 2 a t_{1})$ meet the parabola at $(a t_{2}^{2}, 2 a t_{2})$ then,

$t_{2} = - t_{1} - \frac{2}{t_{1}}$ .........(1)

Here, Given slope of PO= $\frac{2 a t_{1}}{a t_{1}^{2}} = \frac{2}{t_{1}}$

Slope of $Q O = \frac{2 a t_{2}}{a t_{2}^{2}} = \frac{2}{t_{2}}$

And (slope of PO)(slope of QO)=-1

$\Rightarrow (\frac{2}{t_{1}}) (\frac{2}{t_{2}}) = - 1 \Rightarrow t_{1} t_{2} = - 4$ ......(2)

From (1), $t_{2} = - t_{1} - \frac{2}{t_{1}}$

From (2), $t_{2} = - \frac{4}{t_{1}}$

$\Rightarrow \frac{4}{t_{1}} = t_{1} + \frac{2}{t_{1}}$

$4 = t_{1}^{2} + 2$

$\Rightarrow t_{1} = \pm \sqrt{2}$

$\Rightarrow p = (2 a t, a t^{2}) = (2 a (\pm \sqrt{2}), a (2))$

= $(\pm 2 \sqrt{2} a, 2 a)$

$∴$ Only option c satisfies.

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Similar questions

If normal at p to parabola $y^{2} = 4 a x$ meets parabola again at Q such that PQ subtends a right angle at the vertex of $y^{2} = 4 a x$ then p will be

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If normal at p to parabola y2=4ax meets parabola again at Q such that PQ subtends a right angle at the vertex of y2=4ax then p will be

If normal at p to parabola $y^{2} = 4 a x$ meets parabola again at Q such that PQ subtends a right angle at the vertex of $y^{2} = 4 a x$ then p will be