The equation of the circle passing through $(1,0)$ and $(0,1)$ and having the smallest possible radius is:

The equation of the circle which touches the circle $S \equiv x^2+y^2-10x-4y+19=0$ at the point $(2,3)$ internally and has a radius equal to half of the radius of the circle $S=0$ is:

Suppose $S_1$ and $S_2$ are two unequal circles,$AB$ and $CD$ are the direct common tangents to these circles. $A$ transverse common tangent $PQ$ cuts $AB$ in $R$ and $CD$ in $S$. If $AB=10$,then $RS$ is

In the figure,$ABCD$ is a unit square. $A$ circle is drawn with centre $O$ on the extended line $CD$ and passing through $A$. If the diagonal $AC$ is tangent to the circle,then the area of the shaded region is

For the circle $x^2+y^2-9=0$,find the equation of the chord having $(1,2)$ as its mid-point.

The angle between the circles $x^2+y^2-2x-9=0$ and $x^2+y^2-4y-1=0$ at their point of intersection is