Let $A(2,3)$,$B(3,-1)$,and $C(-3,2)$ be three points. If the centre of the circle passing through $A$,$B$,and $C$ is $O(h, k)$,then find the value of $2k - 4h$.

Let the equation of the circle,which touches the $x$-axis at the point $(a, 0), a > 0$ and cuts off an intercept of length $b$ on the $y$-axis,be $x^2 + y^2 - \alpha x + \beta y + \gamma = 0$. If the circle lies below the $x$-axis,then the ordered pair $(2a, b^2)$ is equal to:

Let $R$ be the region of the disc $x^2+y^2 \leq 1$ in the first quadrant. Then,the area of the largest possible circle contained in $R$ is

The equation of the circle which touches the circle $S \equiv x^2+y^2-10x-4y+19=0$ at the point $(2,3)$ internally and has a radius equal to half of the radius of the circle $S=0$ is:

The area of an equilateral triangle inscribed in the circle $x^2+y^2-6x+2y-28=0$ is . . . . . . sq. units.

Two circles $(x + a)^2 + (y + b)^2 = a^2$ and $(x + \alpha)^2 + (y + \beta)^2 = \beta^2$ cut orthogonally if: