If one end of the diameter of $x^2+y^2-2x-6y-15=0$ is $(4,1)$,then the coordinates of the other end are:

Let $ABCD$ be a trapezium with parallel sides $AB$ and $CD$ such that the circle $S$ with $AB$ as its diameter touches $CD$. Further,the circle $S$ passes through the mid-points of the diagonals $AC$ and $BD$ of the trapezium. The smallest angle of the trapezium is

If the diameter of the circle $2(x^2 + y^2) + 3x + 4y - 1 = 0$ is $y = 2x + k$,then $k = \dots$

Let $S_1, S_2,$ and $S_3$ be three circles of unit radius which touch each other externally. The common tangents to each pair of circles are drawn and extended so that they intersect and form a triangle $ABC$ with circumradius $R$. Then $R$ is equal to

The interval of the values of $a$ for which the line $x + y = 0$ bisects $2$ distinct chords drawn from a point $P \left( \frac{1 + \sqrt{2} a}{2}, \frac{1 - \sqrt{2} a}{2} \right)$ to the circle $2x^2 + 2y^2 - (1 + \sqrt{2} a)x - (1 - \sqrt{2} a)y = 0$ 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 $AB$.