Question

The length of the chord of the parabola $x^2=4ay$ passing through the vertex and having slope $\tan \alpha$ is

• A. $4a\cos \text{ec}\alpha \cdot \cot \alpha$
• B. $4a\tan \alpha \cdot \sec \alpha$
• C. $4a\cos \alpha \cdot \cot \alpha$
• D. $4a\sin \alpha \cdot \tan \alpha$

Answer 1

Answer: (B)

Let $A$ be the vertex and $AP$ be a chord of $x^2=4ay$ such that slope of $AP$ is $\tan \alpha$ .
Let the coordinates of $P$ be $(2at,a{t}^2)$ then, Slope of $AP=\frac{a{t}^2}{2at}=\frac{t}{2}$
$\Rightarrow$ $\tan \alpha =\frac{t}{2}$
$\Rightarrow$ $t=2\tan \alpha$ Now, $AP=\sqrt{{(2at-0)}^2+{(a{t}^2-0)}^2}$
$=at\sqrt{4+t^2}=2a\tan \alpha \sqrt{4+4{\tan }^2\alpha }$
$=4a\tan \alpha \cdot \sec \alpha$