If the endpoints of a diameter of a circle are $(1, 2)$ and $(3, 4)$,then its equation is:

The equation obtained by transforming $x^2+y^2-6x+10y-2=0$ to the parallel axes through $(3,-5)$ is

The number of integral values of $K$ for which $x^2+y^2+kx+(1-k)y+5=0$ represents a circle whose radius cannot exceed $5$ is:

The equation of a circle passing through the origin and making an $x$-intercept of $3$ and a $y$-intercept of $-5$ is

The radius of the circle $x^2 + y^2 + 2x \cos \theta + 2y \sin \theta - 8 = 0$ is

The straight line $2x + 3y - k = 0, k > 0$ cuts the $x$ and $y$-axes at $A$ and $B$ respectively. The area of $\triangle OAB$,where $O$ is the origin,is $12 \text{ sq unit}$. The equation of the circle having $AB$ as diameter is