If $\theta$ is the angle between the tangents drawn from the point $(2,3)$ to the circle $x^2+y^2-6x+4y+12=0$,then $\theta=$

Two tangents are drawn from a point $P$ to the circle $x^{2}+y^{2}-2x-4y+4=0$,such that the angle between these tangents is $\tan^{-1}\left(\frac{12}{5}\right)$,where $\tan^{-1}\left(\frac{12}{5}\right) \in (0, \pi)$. If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$,then the ratio of the areas of $\Delta PAB$ and $\Delta CAB$ is:

The line $3x + y - 5 = 0$ touches a circle $S$ at $(1, 2)$. If $(h, k)$ is the centre of the circle $S$ such that $h^2 + hk + k^2 = 37$ and the radius of the circle $S$ is $\sqrt{10}$,then $k =$

The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle $x^2 + y^2 = 4$ and the $x$-axis form a triangle. The area of this triangle (in square units) is

The normal to the circle given by $x^2+y^2-6x+8y-144=0$ at $(8,8)$ meets the circle again at the point

If the line $y = \sqrt{3}x + k$ touches the circle $x^2 + y^2 = 16$,then $k =$