If the equation $\frac{K(x + 1)^2}{3} + \frac{(y + 2)^2}{4} = 1$ represents a circle,then $K = $

For what value of $t$ do the four distinct points $(2, 3), (0, 2), (4, 5),$ and $(0, t)$ lie on a circle?

The radius of a circle with center $(a, b)$ and passing through the center of the circle $x^2 + y^2 - 2gx + f^2 = 0$ is:

$P$ and $Q$ are the points of trisection of the line segment joining the points $(3, -7)$ and $(-5, 3)$. If $PQ$ subtends a right angle at a variable point $R$,then the locus of $R$ is

If the segments of the straight lines $x+y=6$ and $x+2y=4$ are two diameters of a circle passing through $(6,2)$,then the equation of that circle is

Let $S$ be a circle concentric with the circle $3x^2+3y^2+x+y-1=0$. If the length of the tangent drawn from a point $(2,-2)$ to the given circle is the radius of the circle $S$,then the power of the point $(2,1)$ with respect to the circle $S$ is