The normal drawn at $(1,1)$ to the circle $x^2+y^2-4x+6y-4=0$ is

Let a circle $C$ of radius $5$ lie below the $x$-axis. The line $L_{1}: 4x + 3y + 2 = 0$ passes through the centre $P$ of the circle $C$ and intersects the line $L_{2}: 3x - 4y - 11 = 0$ at $Q$. The line $L_{2}$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5x - 12y + 51 = 0$ is

The tangent to the circle $C_1 : x^2 + y^2 - 2x - 1 = 0$ at the point $(2, 1)$ cuts off a chord of length $4$ from a circle $C_2$ whose centre is $(3, -2)$. The radius of $C_2$ is

If the line $3x - 4y = 1$ touches the circle $(x - 1)^2 + (y + 2)^2 = 4$ at $(\alpha, \beta)$,the values of $\alpha$ and $\beta$ are

Two sides of a square are along the lines $x=-5$ and $y=4$. The point of intersection of the diagonals is $(3,-4)$. The point of intersection of the tangents drawn to the circumcircle of the square at the two consecutive vertices lying on $x=-5$ is

Suppose two tangents $PA$ and $PB$ are drawn to the circle centered at $C(1, 2)$ from the point $P(16, 7)$. If the area of the quadrilateral $PACB$ is $75$ square units,then the radius of the circle is: