Question

Write the length of the chord of the parabola y² = 4ax which passes through the vertex and is inclined to the axis at $\frac{\mathrm{\pi }}{4}$.

Answer 1

Let OP be the chord.

Let the coordinates of P be $\left(x_1,y_1\right)$.

From the figure, we have:

$O{P}^2={x_1}^2+{y_1}^2$ (1)

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

$\Rightarrow x_1=y_1$ (2)

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

∴ ${y_1}^2=4a{x}_1$ (3)

Using (2) and (3), we get:
${x_1}^2=4a{x}_1\Rightarrow x_1=4a$ ...(4)

∴ From (4), (1) and (2), we have:
$$\begin{gathered} O{P}^2={\left(4a\right)}^2+{\left(4a\right)}^2=32a^2 \\ \Rightarrow OP=4\sqrt{2}a \end{gathered}$$

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