Suppose a circle passes through $(2,2)$ and $(9,9)$ and touches the $X$-axis at $P$. If $O$ is the origin,then $OP$ is equal to

If the circles $x^2+y^2=9$ and $x^2+y^2+2\alpha x+2y+1=0$ touch each other internally,then the value of $\alpha^3$ is

Let $P_1, P_2, P_3, P_4, P_5$ be five equally spaced points on the circumference of a circle of radius $1$,centred at $O$. Let $R$ be the set of points in the plane of the circle that are closer to $O$ than any of $P_1, P_2, P_3, P_4, P_5$. Then,$R$ is a

If $5x - 12y + 10 = 0$ and $12y - 5x + 16 = 0$ are two tangents to a circle,then the radius of the circle is

Lines are drawn from a point $P(-1, 3)$ to a circle $x^2 + y^2 - 2x + 4y - 8 = 0$. If the line meets the circle at two points $A$ and $B$,then the minimum value of $PA + PB$ is:

Let $C$ be the centre of the circle $x^{2}+y^{2}-x+2 y=\frac{11}{4}$ and $P$ be a point on the circle. $A$ line passes through the point $C$,makes an angle of $\frac{\pi}{4}$ with the line $CP$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $PQR$ (in unit$^{2}$) is.