Consider curves `S_(1): sqrt(|x|)+sqrt(|y|)=sqrt(a), S_(2): x^(2)+y^(2)=a^(2) and S_(3)": "|x|+|y|=a." If "alpha" is area bounded by "S_(1) and S_(2), beta" is area bounded by "S_(1) and S_(3) and gamma` is the area bounded by `S_(2) and S_(3),` then

Options

A. `alpha=a^(2)(pi-(2)/(3))`
B. `beta=(4a^(2))/(3)`
C. `gamma=2a^(2)(pi-1)`
D. the ratio in which `S_(3)` divides area between `S_(1) and S_(2)" is "4:3(pi-2)`

Answer

A, B, D

Explanation

`sqrt(|x|)+sqrt|y|=sqrt(a)`
Consider `x,ygt0`
`therefore" "sqrt(x)+sqrt(y)=sqrt(a)`
`therefore" "y=(sqrt(a)-sqrt(x))^(2)`
`y'=2(sqrt(a)-sqrt(x))(-(1)/(sqrt(x)))=2(-(sqrt(a))/(sqrt(x)))`
`therefore" "y''=(sqrt(a))/(x^(3//2))gt0`
So, graph is concave upward.
Area bounded by `S_(1)` and coordinate axis in first quadrant
`=overset(a)underset(0)int(sqrt(a)-sqrt(x))^(2)dx`
`=overset(a)underset(0)int[a+x-2sqrt(a)sqrt(x)]dx`
`=[ax+(x^(2))/(2)-4sqrt(a)(x^(3//2))/(3)]_(0)^(a)`
`=a^(2)+(a^(2))/(2)-(4a^(2))/(3)`
`=(a^(2))/(6)`
Area bounded by `S_(2)` and coordinate axis in first quadrant `=(pia^(2))/(4)`
Area bounded by `S_(3)` and coordinate axis in first quadrant `=(a^(2))/(2)`
`therefore" Area bounded by "S_(1) and S_(3), alpha=4((pia^(2))/(4)-(a^(2))/(6))`
`=a^(2)(pi-(2)/(3))`
`"Area bounded by "S_(1) and S_(3), beta=4((a^(2))/(2)-(a^(2))/(6))=(4a^(2))/(3)`
`"Area bounded by "S_(2) and S_(3),gamma=4((pia^(2))/(4)-(a^(2))/(2))=a^(2)(pi-2)`
`"Also, "(beta)/(gamma)=(4)/(3sqrt((pi-2)))`