A circle passing through origin cuts the coordinate axes is $A$ and $B$. If the straight line $A B$ passes through a fixed point $\left(x_1, y_1\right)$, then the locus of the centre of the circle is

If $(\alpha, \beta)$ is the external centre of similitude of the circles $x^2+y^2=3$ and $x^2+y^2-2 x+4 y+4=0$, then $\frac{\beta}{\alpha}=$

The equation of the circle touching the lines $|x-2|+|y-3|=4$ is

If the chord joining the points $(1,2)$ and $(2,-1)$ on a circle subtends an angle of $\frac{\pi}{4}$ at any point on its circumference, then the equation of such a circle is