If the points of intersection of the ellipses $x^{2}+2y^{2}-6x-12y+23=0$ and $4x^{2}+2y^{2}-20x-12y+35=0$ lie on a circle of radius $r$ and centre $(a, b)$, then the value of $ab+18r^{2}$ is:

The condition for the coaxial system $x^2+y^2+2 \lambda x+c=0$,where $\lambda$ is a parameter and $c$ is a constant,to have distinct limiting points,is

If the circle $x^2+y^2+6x-2y+k=0$ bisects the circumference of the circle $x^2+y^2+2x-6y-15=0$,then $k$ is equal to :

If the circles $x^2+y^2-2 \lambda x-2 y-7=0$ and $3(x^2+y^2)-8 x+29 y=0$ are orthogonal,then $\lambda=$

If the circles $x^{2}+y^{2}+6x+8y+16=0$ and $x^{2}+y^{2}+2(3-\sqrt{3})x+2(4-\sqrt{6})y = k+6\sqrt{3}+8\sqrt{6}$ with $k>0$ touch internally at the point $P(\alpha, \beta)$,then $(\alpha+\sqrt{3})^{2}+(\beta+\sqrt{6})^{2}$ is equal to $\dots\dots$

Find the radical center of the three circles $x^2 + y^2 = a^2$,$(x - c)^2 + y^2 = a^2$,and $x^2 + (y - b)^2 = a^2$.