Write the length of the chord of the parabola `y^2=4a x` which passes through the vertex and in inclined to the axis at `pi/4` .

Explanation

The vertex and axis of the parabola `y^2 = 4ax` is `(0, 0)` and `y = 0` (x - axis) respectively.

The equation of the straight line passing through the origin and inclines to the x - axis at an angle `theta` is `y = tan theta x`.

`y = tan( frac{pi} {4})x`

`y = 1.x`

`y = x`.

The equation of the chord is `y = x`. Substituting `y = x` in the equation of parabola.

`x^2 = 4ax ⇒ x = 4a. = y = x = 4a` The chord passes through the points `(0, 0)` and `(4a, 4a)`.

The distance between the two points `(x_1, y_1)` and `(x_2, y_2)` is `sqrt(( x_1 − x_2 ) ^2 + ( y_1 − y_2 ) ^2 (x_1−x_2)^2+(y_1−y)^2)`
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