For the circle $x-2=5 \cos \theta, y+1=5 \sin \theta$ where $\theta$ is the parameter,the line $x=1+\frac{r}{2}, y=-2+\frac{\sqrt{3}}{2} r$ where $r$ is the parameter,is a

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.

Two circles in the first quadrant of radii $r_1$ and $r_2$ touch the coordinate axes. Each of them cuts off an intercept of $2$ units with the line $x+y=2$. Then $r_1^2+r_2^2-r_1 r_2$ is equal to $...........$

If $(a, b)$ is the midpoint of the chord $2x - y + 3 = 0$ of the circle $x^2 + y^2 + 6x - 4y + 4 = 0$,then $2a + 3b =$

If a unit circle $S \equiv x^2+y^2+2gx+2fy+c=0$ touches the circle $S^{\prime} \equiv x^2+y^2-6x+6y+2=0$ externally at the point $P(-1, -3)$,then $g+f+c=$

If in two circles,arcs of the same length subtend angles $30^{\circ}$ and $78^{\circ}$ at the centre,then the ratio of their radii is