The length of the tangent drawn from the point $(2, 3)$ to the circle $2(x^2 + y^2) - 7x + 9y - 11 = 0$ is:

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the midpoint of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$,then the radius of the circle $C$ is

If $C(\alpha, \beta)$ with $\alpha < 0$ is the centre of the circle that touches the $Y$-axis at $(0, 3)$ and makes an intercept of length $2$ units on the positive $X$-axis,then $(\alpha, \beta) =$

$A$ rectangle $R$ with end points of one of its sides as $(1, 2)$ and $(3, 6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x - y + 4 = 0$,then the area of $R$ is

$x^2+y^2+2x-6y-6=0$ and $x^2+y^2-6x-2y+k=0$ are two intersecting circles and $k$ is not an integer. If $\theta$ is the angle between the two circles and $\cos \theta = \frac{-5}{24}$,then $k=$

If the foot of the normal from the point $(4,3)$ to a circle is $(2,1)$ and $2x-y-2=0$ is a diameter of the circle,then the equation of the circle is