Question

Write the length of the chord of the parabola y² = 4ax which passes through the vertex and is inclined to the axis at π/4.

Answer 1

Given that,

We need to find the length of the chord of the parabola y² = 4ax which passes through the vertex and is inclined to axis at π/4. 

The figure for the parabola is as follows :

We know that,

The vertex and axis of the parabola y² = 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 = 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. 

⇒ x² = 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 (x₁, y₁) and (x₂, y₂) is \(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\).

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