Question

If a tangent to the parabola $y^2=4ax$ makes an angle of $\frac{\pi }{3}$ with the axis of symmetry of the parabola, then point of contact(s) is/are

• A. $(\frac{a}{3},-\frac{2a}{\sqrt{3}})$
• B. $(3a,-2\sqrt{3}a)$
• C. $(3a,2\sqrt{3}a)$
• D. $(\frac{a}{3},\frac{2a}{\sqrt{3}})$

Answer 1

Answer: (A), (D)

The correct options are
A $(\frac{a}{3},-\frac{2a}{\sqrt{3}})$
D $(\frac{a}{3},\frac{2a}{\sqrt{3}})$

The axis of symmetry of the parabola $y^2=4ax$ is $x-$axis
Any line which makes an angle of $\frac{\pi }{3}$ have slope
$m=\pm \tan \frac{\pi }{3}=\pm \sqrt{3}$
The point of contact in slope form to parabola $y^2=4ax$ is
$\Rightarrow (\frac{a}{m^2},\frac{2a}{m})=(\frac{a}{3},\pm \frac{2a}{\sqrt{3}})$