The least area of a circle circumscribing any right triangle of area `9/pi` is ___________

Answer

9

Explanation

`(9)/(pi)=S=(xy)/(2)`=constant
Area of the circles `A(x)=pir^(2)=pi(x^(2)+y^(2))/(4),(x^(2)+y^(2)=4r^(2))`
`=(pi)/(4)[x^(2)+((2s)/(x))^(2)]`
`A(x)=(pix)/(2)-(2piS^(2))/(x^(3))=0`
or `x^(4)=4S^(2)`
or `S^(2)=(x^(2)y^(2))/(4)=(2Sy^(2))/(4)`
or `y^(2)=2S`
Therefore least area of circle =`pir^(2)=(pi)/(4)(X^(2)+y^(2))`
`=pi`=9 sq.units