Question

Let $\alpha $ be the angle which a tangent to the parabola $y^{2}=4ax$ makes with its axis, the distance between the tangent and a parallel normal will be

• A. $asin^{2}\alpha cos^{2}\alpha $
• B. $acosec\alpha sec^{2}\alpha $
• C. $atan^{2}\alpha $
• D. $acos^{2}\alpha $

Answer 1

Answer: (B)

Tangent of parabola $y^{2}=4ax$ is
$y=mx+\frac{a}{m}$ $.......\left(\right.i\left.\right)$
Where $m=tan\alpha $
And equation of normal parallel to eq. $\left(\right.i\left.\right)$ is
$y=mx-2am-am^{3}$ $....\left(\right.ii\left.\right)$
Distance between eq. $\left(\right.i\left.\right)$ and $\left(\right.ii\left.\right)$ is $\frac{\left|\frac{a}{m} + 2 a m + a m^{3}\right|}{\sqrt{\left(1 + m^{2}\right)}}$
$=\frac{a \left(1 + m^{2}\right)^{2}}{m \sqrt{\left(1 + m^{2}\right)}}=\frac{a \left(1 + m^{2}\right)^{3 / 2}}{m}=\frac{a \left(1 + \left(tan\right)^{2} \alpha \right)^{3 / 2}}{tan \alpha }=\frac{a \left(sec\right)^{3} \alpha }{tan \alpha }$
$=asec^{2}\alpha cosec\alpha $