If the length of the tangent drawn from the point $(-2, 3)$ to the circle $x^2 + y^2 + 8x - 6y + k = 0$ is $4$ units,then $k$ is equal to

Let $C_1$ and $C_2$ be two circles touching each other externally at point $A$. Let $AB$ be the diameter of circle $C_1$. Draw a secant $BA_3$ to circle $C_2$,intersecting circle $C_1$ at point $A_1$ (where $A_1 \neq A$) and circle $C_2$ at points $A_2$ and $A_3$. If $BA_1 = 2$,$BA_2 = 3$,and $BA_3 = 4$,then the radii of circles $C_1$ and $C_2$ are respectively:

Tangent $L_1 \equiv 3x - 4y - 8 = 0$ and the chord $L_2 \equiv x + y - 1 = 0$ are at a distance of $2$ and $\sqrt{2}$ units respectively from the centre of a circle $S$. $(h, k)$ is the centre of $S$ such that $h^2 + k^2 = 13$. If the midpoint of the chord $L_2 = 0$ is $(\alpha, \beta)$ and the radius of the circle is $r$,then $\alpha + \beta + r =$

Let a circle $C$ in the complex plane pass through the points $z_{1}=3+4i$,$z_{2}=4+3i$,and $z_{3}=5i$. If $z(\neq z_{1})$ is a point on $C$ such that the line through $z$ and $z_{1}$ is perpendicular to the line through $z_{2}$ and $z_{3}$,then $\arg(z)$ is equal to

Let a circle $C: (x-h)^{2} + (y-k)^{2} = r^{2}, k > 0$,touch the $x$-axis at $(1, 0)$. If the line $x + y = 0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is $2$,then the value of $h + k + r$ is equal to

If the lines $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ are tangents to the same circle,then the area of that circle (in sq. units) is