The maximum distance of the point $P(10, 7)$ from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ is:

$A$ circle with center at $(2,4)$ is such that the line $x+y+2=0$ cuts a chord of length $6$. The radius of the circle is

If the lengths of the tangents drawn from the point $(1,2)$ to the circles $x^2+y^2+x+y-4=0$ and $3x^2+3y^2-x-y-\lambda=0$ are in the ratio $3:4$,then $\lambda$ is equal to

The equations of the circles,which touch both the axes and the line $4x+3y=12$ and have centers in the first quadrant,are

Let $S=0$ be the circle passing through the points $(2,0)$,$(1,-2)$,and $(-1,1)$. Then the point $(1,2)$

Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8x-6y-23=0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$,then the length of the line segment $AC$ is.