If a tangent to a parabola y2-4ax makes an angle of -3 with the axis of the parabola- Then point of contacts is-are

The axis of the parabola y^2=4ax is x axis any line which makes an angle of π3 will have slope as nbsp;m=±tanπ3=±√(3) ∴ y=±√(3)x±a√(3) point of contact of the ...

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Byju's Answer

Standard XII

Mathematics

Slope Form a Line

If a tangent ...

Question

If a tangent to a parabola $y^{2} = 4 a x$ makes an angle of $\frac{π}{3}$ with the axis of the parabola. Then point of contact(s) is/are

(\frac{a}{3}, - \frac{2 a}{\sqrt{3}})

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(3 a, - 2 \sqrt{3} a)

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(3 a, 2 \sqrt{3} a)

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(\frac{a}{3}, \frac{2 a}{\sqrt{3}})

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Solution

The correct option is D $(\frac{a}{3}, \frac{2 a}{\sqrt{3}})$
The axis of the parabola $y^{2} = 4 a x$ is $x -$ axis
any line which makes an angle of $\frac{π}{3}$ will have slope as $m = \pm tan \frac{π}{3} = \pm \sqrt{3}$
$∴ y = \pm \sqrt{3} x \pm \frac{a}{\sqrt{3}}$
point of contact of the tangent with the parabola can be obtained by
${(\sqrt{3} x + \frac{a}{\sqrt{3}})}^{2} = 4 a x \Rightarrow 3 x^{2} + \frac{a^{2}}{3} + 2 a x = 4 a x$
$\Rightarrow 9 x^{2} - 6 a x + a^{2} = 0 \Rightarrow (3 x - a)^{2} = 0$
$∴ x = \frac{a}{3}$ and $y = \pm \frac{2 a}{\sqrt{3}}$

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If a tangent to a parabola y2=4ax makes an angle of π3 with the axis of the parabola. Then point of contact(s) is/are

If a tangent to a parabola $y^{2} = 4 a x$ makes an angle of $\frac{π}{3}$ with the axis of the parabola. Then point of contact(s) is/are