Let the circles $C_1 : |z| = r$ and $C_2 : |z - 3 - 4i| = 5, z \in \mathbb{C}$, be such that $C_2$ lies within $C_1$. If $z_1$ moves on $C_1, z_2$ moves on $C_2$ and $\min |z_1 - z_2| = 2$, then $\max |z_1 - z_2|$ is equal to:

Tangents are drawn from the point $(17, 7)$ to the circle $x^2 + y^2 = 169$.
Statement-$1$: These tangents are perpendicular to each other.
Statement-$2$: From every point on the circle $x^2 + y^2 = 338$,perpendicular tangents can be drawn to the given circle.

The chord of contact of the tangents drawn from a point on the circle $x^2 + y^2 = a^2$ to the circle $x^2 + y^2 = b^2$ touches the circle $x^2 + y^2 = c^2$. Then $a, b, c$ are in:

Let $AB$ be a chord of the circle $x^2 + y^2 = r^2$ subtending a right angle at the centre. Then the locus of the centroid of the $\Delta PAB$ as $P$ moves on the circle is

$P$ 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

The tangent at $A(-1, 2)$ on the circle $x^2+y^2-4x-8y+7=0$ touches the circle $x^2+y^2+4x+6y=0$ at $B$. Then,a point of trisection of $AB$ is