The angle between the tangents drawn from the point 1 , 4 to the parabola y 2 = 4 x is

Question

Aicharya · Accepted Answer

As we know that any tangent to the parabola $y^{2} = 4 x$ is $y = m x + \frac{1}{m}$

$∵$ Tangents are passing through the points $(1, 4)$

$\Rightarrow 4 = m + \frac{1}{m}$

$\Rightarrow m^{2} - 4 m + 1 = 0$

Let the roots be $m_{1}$ and $m_{2}$

$\Rightarrow m_{1} + m_{2} = 4$ and $m_{1} m_{2} = 1$

Now, ${(m_{1} - m_{2})}^{2} = {(m_{1} + m_{2})}^{2} - 4 m_{1} m_{2} = 16 - 4 = 12$

$\Rightarrow m_{1} - m_{2} = 2 \sqrt{3}$

Now, angle between tangents is $\tan θ = |\frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}|$

$\Rightarrow \tan θ = |\frac{2 \sqrt{3}}{2}| = \sqrt{3}$

$\Rightarrow θ = \frac{π}{3}$

Solution