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

The equation of the circle passing through the point $(-1, -3)$ and touching the line $4x + 3y - 12 = 0$ at the point $(3, 0)$ is

The equation of the tangent to the circle $x^2 + y^2 = a^2$ which makes a triangle of area $a^2$ with the coordinate axes is:

The equation of the normal to the curve $x^{2}+y^{2}=r^{2}$ at the point $P(r \cos \theta, r \sin \theta)$ is:

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 point on the curve ${y^2} = 2(x - 3)$ at which the normal is parallel to the line $y - 2x + 1 = 0$ is