If the circle $x^2+y^2-6x+2y=28$ cuts off a chord of length $\lambda$ units on the line $2x-5y+18=0$,then the value of $\lambda$ is

Let $ABCD$ be a quadrilateral with area $18$,with side $AB$ parallel to the side $CD$ and $AB = 2CD$. Let $AD$ be perpendicular to $AB$ and $CD$. If a circle is drawn inside the quadrilateral $ABCD$ touching all the sides,then its radius is

Let $\alpha$ be an integer multiple of $8$. If $S$ is the set of all possible values of $\alpha$ such that the line $6 x + 8 y + \alpha = 0$ intersects the circle $x^2 + y^2 - 4 x - 6 y + 9 = 0$ at two distinct points,then the number of elements in $S$ is

$AB$ is a line segment of length $24 \ cm$ and $C$ is its midpoint. On $AB$,two semicircles with diameters $AC$ and $CB$ are described on the same side. $A$ large semicircle with diameter $AB$ is also described on the same side. Find the radius of the circle that touches all three semicircles. (in $cm$)

$A$ circle with center at $(2,4)$ is such that the line $x+y+2=0$ cuts a chord of length $6$. The radius of the circle is

The number of circles that touch all the straight lines $x+y=4$,$x-y=-2$,and $y=2$ is