Let a circle of radius 4 pass through the origin O, the points A(-...

Circle of radius 4 passes through O and points A($-\sqrt{3a}$, 0), B(0, $-\sqrt{2b}$).
Let center be C(h, k) on circle: $h^2 + k^2 = 16$.
Since A, B lie on circle: $3a^2 + 2\sqrt{3a}h + h^2 + k^2 = 16$ gives $h = -\frac{\sqrt{3a}}{2}$
And $h^2 + 2b + 2\sqrt{2b}k + k^2 = 16$ gives $k = -\frac{\sqrt{2b}}{2}$
Centroid G = \left(\frac{-\sqrt{3a}}{3}, \frac{-\sqrt{2b}}{3}\right)$. Setting$x = \frac{-\sqrt{3a}}{3}$and$y = \frac{-\sqrt{2b}}{3}$:<br>a = 3x^2 and$b = \frac{9y^2}{2}$<br>From$h^2 + k^2 = 16$:$\frac{3a}{4} + \frac{2b}{4} = 16$→$3a + 2b = 64$<br>Substituting:$9x^2 + 9y^2 = 64$→$x^2 + y^2 = \frac{64}{9}$<br>Radius =$\frac{8}{3}