$ABCD$ is a square with side length $a$. Taking $AB$ and $AD$ as the coordinate axes,find the equation of the circle passing through the vertices of the square.

Two chords of the circle $x^2+y^2-2gx-2hy+g^2+h^2-c^2=0$ pass through the point $(g, h+c)$,and the line $y=x$ bisects these two chords. Then:

The common chord of two intersecting circles $c_1$ and $c_2$ subtends angles of $90^\circ$ and $60^\circ$ at their respective centres. If the distance between their centres is $\sqrt{3} + 1$,then the radii of $c_1$ and $c_2$ are:

Which of the following statements is true for the circle $x^2 + y^2 + 4x - 7y + 12 = 0$?

The equation of the common tangent to the parabola $y^2=8x$ and the circle $x^2+y^2=2$ is $ax+by+2=0$. If $-\frac{a}{b} > 0$,then $3a^2+2b+1=$

Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles,respectively,which pass through the point $(-4, 1)$ and have their centres on the circumference of the circle $x^{2} + y^{2} + 2x + 4y - 4 = 0$. If $\frac{r_{1}}{r_{2}} = a + b \sqrt{2}$,then $a + b$ is equal to: