If the circles $S \equiv x^2+y^2-14x+6y+33=0$ and $S' \equiv x^2+y^2-a^2=0$ where $a \in \mathbb{N}$ have $4$ common tangents,then the possible number of values of $a$ is:

If a point $P$ has coordinates $(0, -2)$ and $Q$ is any point on the circle $x^2 + y^2 - 5x - y + 5 = 0$,then the maximum value of $(PQ)^2$ is

The line $4x - 3y + 15 = 0$ intersects the circle $x^2 + y^2 - 6x - 8y = 0$ at two points $A$ and $B$. The maximum area of $\Delta ABC$,where $C$ is a point on the circumference of the circle,will be - .............. $sq. \ units$.

The equation of the circle circumscribing the triangle formed by the straight lines $x+y=6$,$2x+y=4$,and $x+2y=5$ is given by:

The centre of the circle passing through the point $(0, 1)$ and touching the curve $y = x^2$ at $(2, 4)$ is

Given $r_1, r_2 > 0$ and $C_1, C_2$ are the centres of two circles having only two common tangents. If $C_1 C_2 = r_1 + r_2$,which of the following is correct?