The equation of a circle which touches the $x$-axis and whose centre is $(1, 2)$ is

$A$ line is drawn through a fixed point $P(\alpha, \beta)$ to cut the circle $x^2 + y^2 = r^2$ at $A$ and $B$. Then $PA \cdot PB$ is equal to

If the equation of the circle having its centre in the second quadrant touches the coordinate axes and also the line $\frac{x}{5}+\frac{y}{12}=1$ is $x^2+y^2+2 \lambda x-2 \lambda y+\lambda^2=0$,then $\lambda=$

If the parametric equations of the circle passing through the points $(3,4)$,$(3,2)$,and $(1,4)$ are $x=a+r \cos \theta$ and $y=b+r \sin \theta$,then find the value of $b^{a} r^{a}$.

The radius of the circle with the polar equation $r^2-8r(\sqrt{3} \cos \theta + \sin \theta) + 15 = 0$ is

The equation of a circle of radius $5$ units touching another circle $x^2+y^2-2x-4y-20=0$ at $(5,5)$ is