If a circle passes through the points of intersection of the lines $\lambda x - y + 1 = 0$ and $x - 2y + 3 = 0$ with the coordinate axes,then the value of $\lambda$ is:

Let $AB$ be a chord of a circle and $C$ divides $AB$ internally in the ratio $3 : 1$. $A$ line through $C$ cuts the circle at $D$ and $E$ such that the minimum distances of $D$ and $E$ from line $AB$ are $3$ and $2$ respectively. If $r$ is the minimum length of $AB$ such that $\alpha$ is the angle between $AB$ and $DE$ for this $r$,then the value of '$r\alpha$' 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:

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

Let $A(1, 4)$ and $B(1, -5)$ be two points. Let $P$ be a point on the circle $(x-1)^{2} + (y-1)^{2} = 1$ such that $(PA)^{2} + (PB)^{2}$ has a maximum value. Then the points $P, A,$ and $B$ lie on:

If one end of the diameter of the circle $x^2+y^2-4x-6y+11=0$ is $(3,4)$,then the other end of the diameter is