If the lines joining the origin to the points of intersection of a line $L$ and $x^2+y^2=4$ are the coordinate axes,then the equation of the line $L$ is

In a rectangle $ABCD$,the coordinates of $A$ and $B$ are $(1, 2)$ and $(3, 6)$ respectively,and some diameter of the circumscribing circle of $ABCD$ has the equation $2x - y + 4 = 0$. Then,the area of the rectangle is

Let $C$ be a circle passing through the points $A (2,-1)$ and $B (3,4)$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle $(x-5)^{2}+(y-1)^{2}=\frac{13}{2}$,then $r^{2}$ is equal to

The least distance of the point $A(10, 7)$ from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ is the length of segment $AM$. If $MM'$ is the diameter of the circle,then the lengths of $AM$ and $AM'$ are respectively . . . . . . , . . . . . . units.

Statement $1$: The only circle having radius $\sqrt{10}$ and a diameter along the line $2x + y = 5$ is $x^2 + y^2 - 6x + 2y = 0$.
Statement $2$: $2x + y = 5$ is a normal to the circle $x^2 + y^2 - 6x + 2y = 0$.

The equation of the incircle of the triangle formed by the lines $x=0, y=0$ and $3x+4y-24=0$ is