If the circles $x^2+y^2=9$ and $x^2+y^2-8x-6y+n^2=0$,where $n \in \mathbb{Z}$,have exactly two common tangents,then the number of values for $n$ is

The angle at which the circles $(x - 1)^2 + y^2 = 10$ and $x^2 + (y - 2)^2 = 5$ intersect is

The value of $c$ for which the set,$\{(x, y) | x^2 + y^2 + 2x \le 1 \} \cap \{(x, y) | x - y + c \ge 0\}$ contains only one point in common is :

If two circles of the same radius $a$ and centers at $(2, 3)$ and $(5, 6)$ are orthogonal,find the value of $a$.

$P$ and $Q$ are the ends of a diameter of the circle $x^2+y^2=a^2$ where $a > \frac{1}{\sqrt{2}}$. $s$ and $t$ are the lengths of the perpendiculars drawn from $P$ and $Q$ onto the line $x+y=1$ respectively. When the product $st$ is maximum,the greater value among $s$ and $t$ is

$P$ is a point $(a, b)$ in the first quadrant. If the two circles which pass through $P$ and touch both the coordinate axes cut at right angles,then: