Find the area of the region in the first quadrant enclosed by `x-a xi s ,` the line `x=sqrt(3)\ y` and the circle `x^2+y^2=4.`

Options

A. `pi/3`
B. `pi/6`
C. `pi/3 - (sqrt(3))/(2)`
D. None of these

Answer

A

Explanation

`x^(2) + y^(2) = 4` and `x = sqrt(3) y`
`P(sqrt(3), 1)` = First quadrant

The point of intersection of the line and the circle in the first quardra nt `(sqrt(3) , 1)`.
Area of `DeltaOPA =` Area of `DeltaOPB +` Area of `PAB`
`= 1/2 xxOB xx PB = 1/2 xx sqrt(3) xx 1= (sqrt(3))/(2)`
Area of `PAB int_(sqrt(3))^(2) ydx`
`= int_(sqrt(3))^(2) sqrt(4-x^(2)) dx = [x/2sqrt(4-x^(2)) + 4/2sin^(-1) 'sqrt(3))/(2)]`
`= pi - (sqrt(3))/(2) - (2pi)/(3) = (pi)/(3) - (sqrt(3))/(2)`
Area enclosed by x-axis the line `x = sqrt(3).y` , and th circle `x^(2) + y^(2) = 4` in the first quadrant
`(sqrt(3))/(2) + (pi)/(3) - (sqrt(3))/(2) = (pi)/(3)`