If the line $3x -4y -k = 0 (k > 0)$ touches the circle $x^2 + y^2 -4x -8y -5 = 0$ at $(a, b)$ then $k + a + b$ is equal to :-

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

From the origin chords are drawn to the circle ${(x – 1)^2} + {y^2} = 1$. The equation of the locus of the middle points of these chords is

Let $PQ$ and $RS$ be the tangent at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle, then $(PQ.RS)$ is equal to

If the tangents at the points $P$ and $Q$ on the circle $x ^2+ y ^2-2 x + y =5$ meet at the point $R \left(\frac{9}{4}, 2\right)$, then the area of the triangle $PQR$ is