Two tangents are drawn from the point $( - 2, - 1)$ to the parabola $y^{2} = 4x$. If $\alpha$ is the angle between these tangents, then $ an\alpha\$is equal to

Question

Two tangents are drawn from the point $( - 2, - 1)$ to the parabola $y^{2} = 4x$. If $\alpha$ is the angle between these tangents, then $	an\alpha\$is equal to

Dr. Abubacker · Accepted Answer

(a)
The equation of a tangent to the parabola $y^{2} = 4x$ is $= mx + \frac{1}{m}$ . If it passes through $( - 2, - 1)$, then
$- 1 = - 2m + \frac{1}{m} \Rightarrow \ 2m^{2} - m - 1 = 0$
$m_{1} + m_{2} = \frac{1}{2},\ m_{1} + m_{2} = - \frac{1}{2}$
Now, $	an\alpha = \pm \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}}$
$= \pm \frac{\sqrt{\left( m_{1} + m_{2} ight)^{2} - 4m_{1}m_{2}}}{1 + m_{1}m_{2}}$
$= \pm \frac{\sqrt{1/4 + 4/2}}{1 - 1/2} = 3$

Q. Two tangents are drawn from the point $( - 2, - 1)$ to the parabola $y^{2} = 4x$. If $\alpha$ is the angle between these tangents, then $\tan\alpha\$is equal to

Explanation

Reviewed by Dr. Abubacker

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