If the radical axis of the circles $x^2 + y^2 - 1 = 0$ and $x^2 + y^2 - 2x - 2y + 1 = 0$ forms a triangle of area $A$ with the coordinate axes,then the value of $\frac{1}{A}$ is

If $r_1$ and $r_2$ are the radii of two circles that touch all four circles given by $(x \pm r)^2 + (y \pm r)^2 = r^2$,then find the value of $\frac{r_1+r_2}{r}$.

The shortest distance from the line $3x + 4y = 25$ to the circle $x^2 + y^2 - 6x + 8y = 0$ is

The line $y=mx+c$ intersects the circle $x^2+y^2=r^2$ at two distinct points if:

Let $C$ be the circle $x^2+y^2=1$ in the $XY$-plane. For each $t \geq 0$,let $L_t$ be the line passing through $(0,1)$ and $(t, 0)$. Note that $L_t$ intersects $C$ in two points,one of which is $(0,1)$. Let $Q_t$ be the other point. As $t$ varies between $1$ and $1+\sqrt{2}$,the collection of points $Q_t$ sweeps out an arc on $C$. The angle subtended by this arc at $(0,0)$ is

If the circle $x^2+y^2+2gx+2fy+c=0$ $(c>0)$ touches both the coordinate axes and lies in the third quadrant,then the length of the chord intercepted by the circle on the line $x+y+\sqrt{c}=0$ is