The number of integers in the range of $m$ for which the point $(m, 1)$ lies in the smaller region enclosed by the circle $x^2 + y^2 - 3x + 1 = 0$ and the line $2x - y = 2$ is

If the points $P(3, 1)$ and $Q(6, 5)$ form a triangle with a third point $R(x, y)$ such that the area of $\Delta PQR$ is $6$ square units and $\angle PRQ = \frac{\pi}{2}$,then the number of possible positions for point $R$ is:

$A$ line passing through the point $P(3, 11)$ intersects the circle $x^{2} + y^{2} = 9$ at points $A$ and $B$. Then $PA \cdot PB = . . . . .$

Let $S$ be a subset of the plane defined by $S = \{(x, y) : |x| + 2|y| = 1\}$. Then,the radius of the smallest circle with centre at the origin and having non-empty intersection with $S$ is

If a unit circle $S \equiv x^2+y^2+2gx+2fy+c=0$ touches the circle $S^{\prime} \equiv x^2+y^2-6x+6y+2=0$ externally at the point $P(-1, -3)$,then $g+f+c=$

If the four distinct points $(4,6), (-1,5), (0,0)$ and $(k, 3k)$ lie on a circle of radius $r$,then $10k + r^2$ is equal to