The area of the region `{(x,y):x^(2)+y^(2)le5,||x|-|y||ge1` is

Options

A. `4(pi-tan^(-1)((24)/(7)))-4`
B. `5(pi-tan^(-1)((24)/(7)))-4`
C. `3(pi-tan^(-1)((24)/(7)))-4`
D. `2(pi-tan^(-1)((24)/(7)))-1`

Answer

B

Explanation

Graph of the inequalities is as shown in the following figure

Solving `y=x-1` with the circle `x^(2)+y^(2)=5` we get
`x^(2)+(x-1)^(2)=5 or x=2`
Required area
`=8((1)/(2).1.1+int_(2)^(sqrt5)sqrt(5-x^(2))dx)`
`=4+8((x)/(2)sqrt(5-x^(2))+(5)/(2)sin^(-1)((x)/(sqrt5)))_(7)^(sqrt5)`
`=4+8(0+(5pi)/(4)-1-(5)/(2)sin^(-1)((2)/(sqrt5)))^(2)`
`=10pi-4-20sin^(-1)((2)/(sqrt5))`
`=20((pi)/(2)-sin^(-1)((2)/(sqrt5)))-4`
`=20cos^(-1)((2)/(sqrt5))-4`
`=20tan^(-1)((1)/(2))-4`
`=5(pi-tan^(-1)((24)/(7)))-4(because 4 tan^(-1)((1)/(2))=pi-tan^(-1)((24)/(7)))`