If the points $(2, 0), (0, 1), (4, 5)$ and $(0, c)$ are concyclic,then $c$ is equal to

The equation of a circle passing through the point $(2,8)$,touching the lines $4x-3y-24=0$ and $4x+3y-42=0$,and having the $x$-coordinate of its centre less than or equal to $8$ is

The equation of the chord of the circle $x^2+y^2-4x-10y+25=0$ having its midpoint at $(1,2)$ is

Two parallel chords of a circle of radius $2$ are at a distance $\sqrt{3}+1$ apart. If the chords subtend at the centre,angles of $\frac{\pi}{k}$ and $\frac{2 \pi}{k}$,where $k>0$,then the value of $[k]$ is [Note : $[k]$ denotes the largest integer less than or equal to $k$ ].

The number of common tangents of the circles given by $x^2 + y^2 - 8x - 2y + 1 = 0$ and $x^2 + y^2 + 6x + 8y = 0$ is

If $\sin ^{-1}(a)$ is the acute angle between the curves $x^2+y^2=4x$ and $x^2+y^2=8$ at the point $(2,2)$,then $a$ is equal to