Area bounded by the circle which is concentric with the ellipse `(x^(2))/(25)+(y^(2))/(9) =1` and which passes through `(4,-(9)/(5))`, the vertical chord common to both circle and ellipse on the positive side of x-axis is

Options

A. `(481)/(25)tan^(-1)((9)/(20))-(36)/(5)`
B. `2 tan^(-1)((9)/(20))`
C. `(481)/(25)tan^(-1)((9)/(20))`
D. none of these

Answer

A

Explanation

As eccentricity of ellipse `= (4)/(3)`
Coordinate of foci `= (4,0), (-4,0)`
`rArr (4,-(9)/(5))` is one of the end point of latus-rectum

`rArr` Required area is
`(1)/(2pi) xx pi xx (4^(2) + (9^(2))/(5^(2))) xx 2 tan^(-1) ((9)/(20)) -` area of `Delta POQ`
`= (481)/(25) tan^(-1) ((9)/(20)) -(1)/(2) xx 4xx ((18)/(5))`
`= (481)/(25) tan^(-1) ((9)/(20)) -(36)/(5)`