The length of the chord intercepted by the circle $x^2+y^2-4x+4y+3=0$ on the line $x=3y+13$ is units.

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

The sum of the squares of the lengths of the chords intercepted on the circle,$x^2 + y^2 = 16$,by the lines,$x + y = n$,$n \in N$,where $N$ is the set of all natural numbers,is

Let $AB$ be a line segment with mid-point $C$ and $D$ be the mid-point of $AC$. Let $C_1$ be the circle with diameter $AB$ and $C_2$ be the circle with diameter $AC$. Let $E$ be a point on $C_1$ such that $EC$ is perpendicular to $AB$. Let $F$ be a point on $C_2$ such that $DF$ is perpendicular to $AB$ and $E$ and $F$ lie on opposite sides of $AB$. Then,the value of $\sin \angle FEC$ is

The midpoint of the chord of the circle $x^2 + y^2 = 25$ intercepted by the line $x - 2y = 2$ is

Consider a circle $C$ which touches the $y$-axis at $(0,6)$ and cuts off an intercept $6 \sqrt{5}$ on the $x$-axis. Then the radius of the circle $C$ is equal to: