The area of the quadrilateral formed by the tangents from the point $(4,5)$ to the circle $x^2+y^2-4x-2y-11=0$,and the pair of radii joining the points of contact of these tangents is

The point $(5, -7)$ lies outside the circle:

In a circle with center $O$,suppose $A, P, B$ are three points on its circumference such that $P$ is the mid-point of the minor arc $AB$. Suppose when $\angle AOB = \theta$,$\frac{\text{area}(\triangle AOB)}{\text{area}(\triangle APB)} = \sqrt{5} + 2$. If $\angle AOB$ is doubled to $2\theta$,then the ratio $\frac{\text{area}(\triangle AOB)}{\text{area}(\triangle APB)}$ is

The points of intersection of the line $4x - 3y - 10 = 0$ and the circle $x^2 + y^2 - 2x + 4y - 20 = 0$ are

The number of tangents that can be drawn from $(0, 0)$ to the circle $x^2 + y^2 + 2x + 6y - 15 = 0$ is

The area of the curve $x^2 + y^2 = 2ax$ is