If the circles $(x+a)^2+(y+b)^2=a^2$ and $(x+c)^2+(y+d)^2=d^2$ cut orthogonally,then $b(b-2d) =$

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

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

Consider a triangle $\Delta$ whose two sides lie on the $x$-axis and the line $x+y+1=0$. If the orthocenter of $\Delta$ is $(1,1)$,then the equation of the circle passing through the vertices of the triangle $\Delta$ is

The length of the chord intercepted by the circle $x^2+y^2-8x-2y-8=0$ on the line $x+y+1=0$ is:

The number of possible common tangents that can be drawn to the circles $x^2+y^2+4x-6y-3=0$ and $x^2+y^2+4x-2y+1=0$ is