Question

The point of the contact of the tangent to the parabola $y^2 = 4ax$ which makes an angle of $30^{\circ}$ with $X$-axis, is

• A. $\left(3a, 2\sqrt{3}a\right)$
• B. $\left( 2\sqrt{3}a, 3a\right)$
• C. $\left( \sqrt{3}a, 6a\right)$
• D. None of these

Answer 1

Answer: (A)

We have, slope of tangent $=\tan 30^{\circ}=\frac{1}{\sqrt{3}}$
Now, the equation of tangent at $(h, k)$ to $y^{2}=4 a x$ is $y k=2 a(x+h)$
On comparing the slopes, we get
$\frac{2 a}{k} =\frac{1}{\sqrt{3}} $
or $k =2 \sqrt{3} a$
and $h =3 a $
$\therefore \, (h, k)=(3 a, 2 \sqrt{3 a})$