The point at which the normal to the circle $x^2 + y^2 + 4x + 6y - 39 = 0$ at the point $(2, 3)$ meets the circle again is:

The area (in sq units) of the triangle formed by the tangent,normal at $(1, \sqrt{3})$ to the circle $x^2+y^2=4$ and the $X$-axis,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 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 combined equation of the tangent and normal to the curve $xy = 100$ at the point $(5, 20)$ is . . . . . . .

$x-2y-6=0$ is a normal to the circle $x^2+y^2+2gx+2fy-8=0$. If the line $y=2$ touches this circle,then the radius of the circle can be