The equation of a circle that passes through the origin and cuts off intercepts $-2$ and $3$ on the $X$-axis and $Y$-axis respectively is

Let the abscissae of the two points $P$ and $Q$ be the roots of $2x^{2}-rx+p=0$ and the ordinates of $P$ and $Q$ be the roots of $y^{2}-sy-q=0$. If the equation of the circle described on $PQ$ as diameter is $2(x^{2}+y^{2})-11x-14y-22=0$,then $2r+s-2q+p$ is equal to

If a circle passing through the point $(-1, 0)$ touches the $y-$ axis at $(0, 2)$,then the length of the chord of the circle along the $x-$ axis is

The sides of a rectangle are given by the equations $x=-2, x=4, y=-2$ and $y=5$. Then the equation of the circle,whose centre is the point of intersection of the diagonals,lying within the rectangle and touching only two opposite sides,is

Let the circle $S = x^2 + y^2 + 2gx + 2fy + c = 0$ touch the positive $X$-axis and the positive $Y$-axis. Let $(2, 4)$ be a point on the circle $S = 0$. If two such circles exist, then the difference of their areas is (in $\pi$)

The circle $x^2+y^2-8x-12y+\alpha=0$ lies in the first quadrant without touching the coordinate axes. If $(6, 6)$ is an interior point to the circle,then