The equation of the circle passing through $(2,0)$ and $(0,4)$ and having the minimum radius is:

If the circles $x^2 + y^2 = a^2$ and $x^2 + y^2 - 2gx + g^2 - b^2 = 0$ touch each other externally,then:

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

$A$ cow is tied to a post by a rope. The cow moves along a circular path,always keeping the rope tight. If it describes an arc of $44 \ m$ when it has traced out $72^{\circ}$ at the centre,the length of the rope is: (in $m$)

For each natural number $k$,let $C_k$ denote the circle with radius $k$ centimeters and center at the origin. On the circle $C_k$,a particle moves $k$ centimeters in the counter-clockwise direction. After completing its motion on $C_k$,the particle moves to $C_{k+1}$ in the radial direction. The motion of the particle continues in this manner. The particle starts at $(1, 0)$. If the particle crosses the positive direction of the $x$-axis for the first time on the circle $C_n$,then $n$ is equal to

The equation of the circle which passes through $(1, 0)$ and $(0, 1)$ and has its radius as small as possible,is