Question

The equation of a tangent to the parabola, $x^2=8y$, which makes an angle $\theta$ with the positive direction of $x$-axis, is :

• A. $y=x\tan \theta -2\cot \theta$
• B. $x=y\cot \theta +2\tan \theta$
• C. $x=y\cot \theta -2\tan \theta$
• D. $y=x\tan \theta +2\cot \theta$

Answer 1

The correct option is B $x=y\cot \theta +2\tan \theta$

$x^2=8y\Rightarrow \frac{dy}{dx}=\frac{x}{4}=\tan \theta$
$\therefore x=4\tan \theta$
and $y=2{\tan }^2\theta$

Equation of tangent at $(4\tan \theta ,2{\tan }^2\theta )$ is :
​​​​$y-2{\tan }^2\theta =\tan \theta (x-4\tan \theta )$
or, $x=y\cot \theta +2\tan \theta$