Question

In the parabola $y^2=4ax$, the length of the chord passing through the vertex and inclined to the axis at $\frac{\pi }{4}$ is

• A. $4\sqrt{2}a$
• B. $2\sqrt{2}a$
• C. $\sqrt{2}a$
• D. none of these

Answer 1

The correct option is A

$4\sqrt{2}a$

Let OP be the chord.

Let the coordinates of P be $(x_1,y_1)$

From the figure, we have:

$O{P}^2=x_1^2+y_1^2...\left(1\right)$

And,$tan\frac{\pi }{4}=\frac{y_1}{x_1}$

$\Rightarrow x_1=y_1...\left(2\right)$

Also,$(x_1,y_1)$ lies on the parabola.

$\therefore y_1^2=4a{x}_1...\left(3\right)$

Using (2) and (3) :

$x_1^2=4a{x}_1\Rightarrow x_1=4a...\left(4\right)$

$\therefore \text{From (4),(1) and (2),we have:}$

$O{P}^2=(4a{)}^2+(4a{)}^2=32a^2$

$\Rightarrow OP=4\sqrt{2}a$

Therefore,the length of the chord is $4\sqrt{2}a$ units.