If the circle $x^2+y^2-4x-8y-5=0$ intersects the line $3x-4y-m=0$ in two distinct points,then the number of integral values of '$m$' is

Let the maximum and minimum values of $(\sqrt{8x-x^2-12}-4)^2+(x-7)^2, x \in R$ be $M$ and $m$ respectively. Then $M^2-m^2$ is equal to ...............

The line $y=mx+c$ intersects the circle $x^2+y^2=r^2$ at two distinct points if:

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

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

Tangent $L_1 \equiv 3x - 4y - 8 = 0$ and the chord $L_2 \equiv x + y - 1 = 0$ are at a distance of $2$ and $\sqrt{2}$ units respectively from the centre of a circle $S$. $(h, k)$ is the centre of $S$ such that $h^2 + k^2 = 13$. If the midpoint of the chord $L_2 = 0$ is $(\alpha, \beta)$ and the radius of the circle is $r$,then $\alpha + \beta + r =$