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
$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$
$=\left|\frac{2\sqrt{10}}{1-1}\right|=\infty$
$\Rightarrow$ $\theta =90^{\circ }$