Find the ratio in which the area bounded by the curves `y^2=12 xa n d x^2=12 y` is divided by the line `x=3.`

Options

A. `(245)/(4)`
B. `(147)/(4)`
C. `(45)/(4)`
D. `(137)/(4)`

Answer

b

Explanation

Let `A_(1) and A_(2)` be areas of regions OPRO and PQRS respectively . Then,

`A_(1)=underset(0)overset(3)int (sqrt(12x)-(x^(2))/(12))dx and A_(2) = underset(3)overset(12)int (sqrt(12x)-(x^(2))/(12))dx`
`implies A_(1)=[2sqrt(3)(x^(3//2))/(3//2)-(x^(3))/(36)]_(0)^(3) and A_(2)=[2sqrt(3)(x^(3//2))/(3//2)-(x^(6))/(36)]_(3)^(12)`
`implies A_(1)=(4)/(sqrt(3))xx3sqrt(3)-(27)/(36)`
and , `A_(2)=(4)/(sqrt(3)){(12sqrt(12)-3sqrt(3))}-(1)/(36)(12^(3)-3^(3))`
`implies A_(1)=12-(3)/(4) and A_(2)=(4)/(sqrt(3)){24sqrt(3)-3sqrt(3)}-(189)/(4)=(147)/(4)`
`implies A_(1)=(45)/(4) and A_(2)=(147)/(4)`
Hence, required answer is `(147)/(4)`.