A variable circle having fixed radius 'a' passes through origin and meets the co-ordinate axes in point A and B. Locus of centroid of triangle OAB where 'O' being the origin, is -

Options

A. `9(x^(2)+y^(2)) =4a^(2)`
B. `9(x^(2)+y^(2))=a^(2)`
C. `9(x^(2)+y^(2))=2a^(2)`
D. `9(x^(2)+y^(2))=8a^(2)`

Answer

A

Explanation

Equation of circle
`(x-alpha)(x-0) +(y-0)(y-beta)=0`

`x^(2) +y^(2) -alphax-betay=0`
`a=sqrt((alpha^(2))/(4)+(beta^(2))/(r))`
`alpha^(2) +beta^(2) =4a^(2)`
Let centroid be (x,y)
`x=alpha//3, y= beta//3`
`alpha =3x, beta =3y`
`:. " locus " 9(x^(2)+y^(2)) =4a^(2)`