Find the equation of the director circle of the circle $x^2 + y^2 = 8$.

Let $ABC$ be a triangle with $AB=1$,$AC=3$,and $\angle BAC=\frac{\pi}{2}$. If a circle of radius $r>0$ touches the sides $AB$,$AC$ and also touches the circumcircle of triangle $ABC$ internally,then the value of $r$ is:

Among all the parallelograms whose diagonals are $10$ and $4$,the one having maximum area has its perimeter lying in the interval

If a circle $C$ passing through the point $(4, 0)$ touches the circle $x^2+y^2+4x-6y=12$ externally at the point $(1, -1)$,then the radius of $C$ is

The radius of the circle having $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ as its tangents 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