The number of tangents that can be drawn from $(0, 0)$ to the circle $x^2 + y^2 + 2x + 6y - 15 = 0$ is

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

Let $A(2, 3)$,$B(4, 5)$ and let $C = (x, y)$ be a point such that $(x - 2)(x - 4) + (y - 3)(y - 5) = 0$. If the area of $\Delta ABC = \sqrt{2} \text{ sq. unit}$,then the maximum number of positions of $C$ in the $xy$ plane is:

For the circle $x^2 + y^2 - 2x + 4y - 4 = 0$,what is the line $2x - y - 1 = 0$?

Position of the point $(1, 1)$ with respect to the circle $x^2 + y^2 - x + y - 1 = 0$ is

Let $S_1, S_2,$ and $S_3$ be three circles of unit radius which touch each other externally. The common tangents to each pair of circles are drawn and extended so that they intersect and form a triangle $ABC$ with circumradius $R$. Then $R$ is equal to