Answer :

Given that we need to find the length of the chord of the parabola y^{2} = 4ax which passes through the vertex and is inclined to axis at . The figure for the parabola is as follows:

We know that the vertex and axis of the parabola y^{2} = 4ax is (0, 0) and y = 0(x - axis) respectively.

We know that the equation of the straight line passing through the origin and inclines to the x - axis at an angle θ is y = tanθx.

⇒

⇒ 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).

We know that the distance between the two points (x_{1}, y_{1}) and (x_{2}, y_{2}) is .

⇒

⇒

⇒

⇒

∴The length of the chord is 4√2a units.

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