Question

From the point $ (-1,-6) $ two tangents are drawn to the parabola $ y^{2}=4x $ . Then, the angle between the two tangents is

• A. $ 30^{\circ} $
• B. $ 45^{\circ} $
• C. $ 60^{\circ} $
• D. $ 90^{\circ} $

Answer 1

Answer: (D)

We know that tangent to $ y^{2}=4ax $
is $ y=mx+\frac{a}{m} $
$ \therefore $ Tangent to $ y^{2}=4x $
is $ y=mx+\frac{1}{m} $ .
Since, tangent passes through $ (1-,-6) $ .
$ \therefore $ $ -6=-m+\frac{1}{m} $
$ \Rightarrow $ $ m^{2}-6m-1=0 $
Whose roots are $ m_{1} $ and $ m_{2} $ .
$ \therefore $ $ m_{1}+m_{2}=6 $ and $ m_{1}\,m_{2}=-1 $
Now, $ |m_{1}-m_{2}|=\sqrt{(m_{1}+m_{2})}^{2}-4m_{1}m_{2} $
$ =\sqrt{36+4} $ $ =2\sqrt{10} $
Thus, angle between tangent is
$ \tan \theta =\left| \frac{m_{2}-m_{1}}{1+m_{1}m_{2}} \right| $
$ =\left| \frac{2\sqrt{10}}{1-1} \right|=\infty $
$ \Rightarrow $ $ \theta =90^{\circ} $