If the line $x - y + 1 = 0$ is a chord of the circle $x^2 + y^2 + 4y - 4 = 0$,find the length of this chord.

Let $ABCD$ be a square of side length $1$,and $\Gamma$ be a circle passing through $B$ and $C$,and touching $AD$. The radius of $\Gamma$ is

The tangents drawn from a point $(2,-1)$ touch the circle $x^2+y^2+4x-2y+1=0$ at the points $A$ and $B$. If $C$ is the centre of the circle,then the area (in sq. units) of the triangle $ABC$ is

The curve $xy = c, (c > 0)$,and the circle $x^2 + y^2 = 1$ touch at two points. Then the distance between the points of contact is

The line $2x - 3y = 1$ divides the circular region $x^2 + y^2 \leq 6$ into two parts. If $S = \left\{ \left(2, \frac{3}{4}\right), \left(\frac{5}{2}, \frac{3}{4}\right), \left(\frac{1}{4}, -\frac{1}{4}\right), \left(\frac{1}{8}, \frac{1}{4}\right) \right\}$,then the number of points in the set $S$ that lie inside the smaller part is:

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