In an acute-angled $\triangle ABC$,the altitudes from $A, B, C$ when extended intersect the circumcircle again at points $A_1, B_1, C_1$ respectively. If $\angle ABC = 45^{\circ}$,then $\angle A_1 B_1 C_1$ equals (in $^{\circ}$)

If $\theta$ is the angle between the circles $x^2+y^2-2x-4y-4=0$ and $x^2+y^2-8x-12y+43=0$,then $|7 \sec \theta - 18 \cos \theta| = $

If a line drawn from a fixed point $M(a, b)$ cuts the circle $x^2+y^2=k^2$ at $C$ and $D$,then $MC \times MD$ is equal to

The diameter of the circle,whose center lies on the line $x+y=2$ in the first quadrant and which touches both the lines $x=3$ and $y=2$,is

In a circle of diameter $40 \, cm$,the length of a chord is $20 \, cm$. Find the length of the minor arc of the chord.

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