A circle of radius 'r' passes through the origin `O` and cuts the axes at A and B,Locus of the centroid of triangle OAB is

Options

A. `x^(2)+y^(2)=k^(2)`
B. `x^(2)+y^(2)=2k^(2)`
C. `x^(2)+y^(2)=3k^(2)`
D. none of these

Answer

D

Explanation

Let the coordinates of A and B be (a, 0) and (0, b) respectively . Clearly, `DeltaOAB` is a right-angled triangle, the hypotenuse AB is a diameter of the circle.

`:. AB=2(3k)=6k`.
Now, `OA^(2)+OB^(2)=AB^(2)rArr a^(2)+b^(2)=36k^(2) " " ...(i) `
Let `(alpha, beta)` be the coordinates of the centroid of `Delta OAB`. Then,
`alpha = a//3, beta= b//3 rArr a=3alpha, b=3 beta`.
Substituting the value of a and b in (i), we get
`9 alpha^(2)+9beta^(3)=36k^(2)rArr alpha^(2)+beta^(2)=4k^(2)`
Hence, the locus of `(alpha, beta)` is `x^(2)+y^(2)=4k^(2)`