If `sin^(-1)((sqrt(x))/2)+sin^(-1)(sqrt(1-x/4))+tan^(-1)y=(2pi)/3`, then

Options

A. maximum value of `x^(2)+y^(2)` is `(49)/(3)`
B. maximum value of `x^(2)+y^(2)` is 4
C. minimum value of `x^(2)+y^(2)` is `(1)/(3)`
D. minimum value of `x^(2)+y^(2)` is 3

Answer

A, C, D

Explanation

`sin^(-1)((sqrt(x))/(2))+sin^(-1)(sqrt(1-(x)/(4)))+tan^(-1)y=(2pi)/(3),x in [0,4]`
`therefore sin^(-1)((sqrt(x))/(2))+cos^(-1)((sqrt(x))/(2))+tan^(-1)y=(2pi)/(3)`
`therefore (pi)/(2)+tan^(-1)y=(2pi)/(3)`
`therefore tan^(-1)y=(2pi)/(3)-(pi)/(2)=(pi)/(6)`
`rArr =(1)/(sqrt(3))`
`therefore` maximum value of `(x^(2)+y^(2))=16+(1)/(3)=(49)/(3)`
and minimum value of `(x^(2)+y^(2))=(0)^(2)+(1)/(3)=(1)/(3)`