Calculate the area bouded by the parabola `y^2=4ax` and its latus rectum

Options

A. `5/3 a^2` sq units
B. `8 a^2` sq units
C. `8/3a^2` sq units
D. `8/3 a^2` sq units

Answer

D

Explanation

Let s(a,0) be the foucus of the parabola `y^2 = 4ax `The its latus rectum LSL is the line parallel to the y-axis at a distance a from it
So,its equation is x =a
Since the equation of the parabola contains only even powers of y, it is symmetrical about the x-axis.
`therefore` required area
= area LOL'L
= (area LOSL)+(area SOL's)
`= 2 xx (area LOSL)`
`2underset(0)overset(a)inty dx =2.underset(0)overset(a)int2sqrt(ax)dx`
`=4sqrt(a).2/3[x^(3//2)]_0^a=8/3 a^2` sq units