If a circle $C$ passing through $(4, 0)$ touches the circle $x^2 + y^2 + 4x - 6y - 12 = 0$ externally at a point $(1, -1),$ then the radius of the circle $C$ is

The interval of the values of $a$ for which the line $x + y = 0$ bisects $2$ distinct chords drawn from a point $P \left( \frac{1 + \sqrt{2} a}{2}, \frac{1 - \sqrt{2} a}{2} \right)$ to the circle $2x^2 + 2y^2 - (1 + \sqrt{2} a)x - (1 - \sqrt{2} a)y = 0$ is:

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:

If $(1, a)$ and $(b, 2)$ are conjugate points with respect to the circle $x^2+y^2=25$,then $4a+2b=$

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:

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.