The area of the triangle formed by the positive $x$-axis and the normal and the tangent to the circle $x^2 + y^2 = 4$ at $(1, \sqrt{3})$ is

The line $(x - a)\cos \alpha + (y - b)\sin \alpha = r$ will be a tangent to the circle $(x - a)^2 + (y - b)^2 = r^2$:

$O(0,0)$ and $A(1,0)$ are the centers of two unit circles $C_1$ and $C_2$ respectively. $C_3$ is also a unit circle having its center above the $X$-axis and passing through $O$ and $A$. The equation of the common tangent to $C_1$ and $C_3$ which does not intersect the circle $C_2$ is

If the acute angle between the pair of tangents drawn from the origin to the circle $x^2+y^2-4x-8y+4=0$ is $\alpha$,then $\tan \alpha=$

The angle between the tangents drawn from the point $P(k, 6k)$ to the circle $x^2+y^2+6x-6y+2=0$ is $2 \operatorname{Tan}^{-1}\left(\frac{4}{3}\right)$. If the coordinates of $P$ are integers,then $k=$

The normal at a point $\theta$ to the curve $x=a(1+\cos \theta), y=a \sin \theta$ always passes through the fixed point