Question

If a chord of the parabola $y^2=4x$ passes through its focus and makes an angle $\theta$ the $X-$axis, then its length is

• A. $4{\cos }^2\theta$
• B. $4{\sin }^2\theta$
• C. $4{\text{cosec}}^2\theta$
• D. $4{\sec }^2\theta$

Answer 1

The correct option is C $4{\text{cosec}}^2\theta$

Parabola: $y^2=4x$

Here, $a=1$

Let's assume the extremities of this focal chord are $A\equiv (x_1,y_1)$ and $B\equiv (x_2,y_2)$

According to the definition of parabola, focal distance of a point is equal to the distance of point from directrix.

We need to find $AB=AF+FB$, where $F$ is the focal point $F\equiv (1,0)$

Distance of $A$ from $F$ is equal to distance of $A$ from Directrix $x=-1$.

From figure, we can say that this distance will be equal to

$AF=1+1+AF\cos \theta$

$\Rightarrow AF=\frac{2}{1-\cos \theta }$

Distance of $B$ from $F$ is equal to distance of $B$ from Directrix $x=-1$.

From figure, we can say that this distance will be equal to

$FB=1+1-FB\cos \theta$

$\Rightarrow FB=\frac{2}{1+\cos \theta }$

Now, total distance $AB$ is

$AB=AF+FB=\frac{2}{1-\cos \theta }+\frac{2}{1+\cos \theta }$

$\Rightarrow AB=\frac{4}{1-{\cos }^2\theta }=\frac{4}{{\sin }^2\theta }=4{\text{cosec}}^2\theta$