Let $6$ and $8$ be the $X$ and $Y$-intercepts made by the circle $S \equiv x^2+y^2+2gx+2fy+c=0$ respectively. If $gx+fy+1=0$ is a line passing through the point $(1, -1)$,then the radius of the circle $S=0$ is

$A$ circle $C_1$ of radius $2$ touches both $x$-axis and $y$-axis. Another circle $C_2$ whose radius is greater than $2$ touches circle $C_1$ and both the axes. Then the radius of circle $C_2$ is

The tangents drawn from a point $(2,-1)$ touch the circle $x^2+y^2+4x-2y+1=0$ at the points $A$ and $B$. If $C$ is the centre of the circle,then the area (in sq. units) of the triangle $ABC$ is

The equation of the chord of the circle $x^2 + y^2 = a^2$ having $(x_1, y_1)$ as its mid-point is

The number of common tangents to two circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 8x + 12 = 0$ is

If two parallel chords of a circle,having diameter $4 \, \text{units}$,lie on the opposite sides of the centre and subtend angles $\cos^{-1}\left(\frac{1}{7}\right)$ and $\sec^{-1}(7)$ at the centre respectively,then the distance between these chords is: