What is the area of the ellipse `4x^(2) + 9y^(2) = 1`.

Options

A. `6pi`
B. `(pi)/(36)`
C. `2(e-1)`
D. `e-1`

Answer

C

Explanation

Given `4x^(2) + 9y^(2) = 1`
`rArr (x^(2))/((1/2)^(2)) + (y^(2))/((1/3)^(2)) = 1`
`:. a = 1/2` and `b = 1/3`

Now, area of ellipse `= 4 int_(0)^(1//2) ydx`
`= 4int_(0)^(1//2)sqrt((1-4x^(2))/(9)) dx = 4/3 int_(0)^(1//2)sqrt(1-(2x)^(2)) dx`
`= 2/3int_(0)^(1) sqrt(1-t^(2))dt = 2/3[1/2tsqrt(1-t^(2)) + 1/2sin^(-1)(t)]_(0)^(1)`
`= 2/3[0+1/2sin^(-1) (1) - 1/2 sin^(-1) (0)]`
`= 2/3[1/2 xx pi/2 - 1/2 xx 0] = pi/6`