A circle $C$ touches $X$-axis and makes an intercept of length 2 units on $Y$-axis. If the centre of this circle lies on the line $y=x+1$, then a circle passing through the centre of the circle $C$ is

If $m_1, m_2$ are the slopes of the tangents drawn through the point $(-1,-2)$ to the circle $(x-3)^2+(y-4)^2=4$, then $\sqrt{3}\left|m_1-m_2\right|=$

A line meets the circle $x^2+y^2-4 x-4 y-8=0$ in two points $A$ and $B$. If $P(2,-2)$ is a point on the circle such that $P A=P B=2$, then the equation of the line $A B$ is

If the centre $(\alpha, \beta)$ of a circle cutting the circles $x^2+y^2-2 y-3=0$ and $x^2+y^2+4 x+3=0$ orthogonally lies on the line $2 x-3 y+4=0$, then $2 \alpha+\beta=$