$A$ 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

Let the tangents drawn from $P(-1, -1)$ to the circle $x^2 + y^2 - 2x - 4y - 4 = 0$ touch the circle at the points $A$ and $B$. Then the area of the triangle $PAB$ (in square units) is

If the line $y - 1 = m(x - 1)$ intersects the circle $x^2 + y^2 = 4$ at two distinct points,then the number of possible values of $m$ is:

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

Two circles of radii $4 \text{ cm}$ and $1 \text{ cm}$ touch each other externally and $\theta$ is the angle contained by their direct common tangents. Then $\sin \theta =$

The line $4x - 3y + 15 = 0$ intersects the circle $x^2 + y^2 - 6x - 8y = 0$ at two points $A$ and $B$. The maximum area of $\Delta ABC$,where $C$ is a point on the circumference of the circle,will be - .............. $sq. \ units$.