Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. 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. . . . . .

A tangent to the circle ${x^2} + {y^2} = 5$at the point $(1,-2)$ the circle ${x^2} + {y^2} - 8x + 6y + 20 = 0$

Let $C$ be the circle with centre at $(1, 1)$ and radius $= 1$. If  $T$ is the circle centred at $(0, y),$ passing through origin and touching the circle $C$ externally, then the radius of $T$ is equal 

If the line $3x -4y -k = 0 (k > 0)$ touches the circle $x^2 + y^2 -4x -8y -5 = 0$ at $(a, b)$ then $k + a + b$ is equal to :-

The equations of the tangents to the circle ${x^2} + {y^2} = 13$ at the points whose abscissa is $2$, are

The gradient of the normal at the point $(-2, -3)$ on the circle ${x^2} + {y^2} + 2x + 4y + 3 = 0$ is