Question

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

• A. $4\cos \alpha \cos \text{e}{\text{c}}^2\alpha$
• B. $4\tan \alpha \sec \alpha$
• C. $4\sin \alpha {\sec }^2\alpha$
• D. None of these

Answer 1

Answer: (A)

Let $A$ be the vertex of the parabola and $AP$ is chord of parabola such that slope of $AP$ is $\cot \alpha$ .
Let coordinates of $P$ be $(2t,t^2)$
which is a point on the parabola.
$\therefore$ Slope of $AP=\frac{t}{2}$
$\Rightarrow$ $\cot \alpha =\frac{t}{2}$
$\Rightarrow$ $t=2\cot \alpha$
In $\Delta APB$ , $AP=\sqrt{4t^2+t^4}$
$=t\sqrt{4+t^2}$
$\therefore$ $AP=2\cot \alpha \sqrt{4(1+{\cot }^2\alpha )}$
$=2\cot \alpha \sqrt{4\cos \text{e}{\text{c}}^2\alpha }=4\cot \alpha \cos \text{ec}\alpha$
$=4\frac{\cos \alpha }{\sin \alpha }\cos \text{ec}\alpha =4\cos \alpha \cos \text{e}{\text{c}}^2\alpha$