If the radius of a circle $x^{2}+y^{2}-4x+6y-k=0$ is $5$,then $k=$

If a circle passing through $(4, -2)$ is concentric with the circle $x^2 + y^2 - 2x + 4y + 20 = 0$,then find the value of $c$ for the circle $x^2 + y^2 - 2x + 4y + c = 0$.

If ${g^2} + {f^2} = c$,then the equation ${x^2} + {y^2} + 2gx + 2fy + c = 0$ will represent

The equation of the circle whose centre lies on the lines $3x - y - 4 = 0$ and $x + 3y + 2 = 0$ and has an area of $154$ square units is:

The diameters of a circle are along $2x + y - 7 = 0$ and $x + 3y - 11 = 0$. Then,the equation of this circle,which also passes through $(5, 7)$,is

The equations of the two circles which touch the $Y$-axis at $(0,3)$ and make an intercept of $8$ units on the $X$-axis are