Let $R$ be a rectangle,$C$ be a circle,and $T$ be a triangle in the plane. The maximum possible number of points common to the perimeters of $R, C$ and $T$ is

Let $A_0 A_1 A_2 A_3 A_4 A_5$ be a regular hexagon inscribed in a circle of unit radius. The product of the lengths of the line segments $A_0A_1$,$A_1A_2$,and $A_0A_4$ is

If the number of common tangents to the pair of circles $x^2+y^2-2x+4y-4=0$ and $x^2+y^2+4x-4y+\alpha=0$ is $4$,then the least integral value of $\alpha$ is

If a line passing through the point $\left( -\sqrt{8}, \sqrt{8} \right)$ and making an angle of $135^{\circ}$ with the $x$-axis intersects the circle $x = 5 \cos \theta, y = 5 \sin \theta$ at points $A$ and $B$,then find the length of the chord $AB$.

The equation of a circle touching the curve $|x - 1| + |y - 4| = 6$ is

The angle subtended by the chord $x+y-1=0$ of the circle $x^2+y^2=1$ at the origin is