Two ships leave a port at the same time. One of them moves in the direction of $E 50^{\circ} N$ with a speed of $8 \text{ kmph}$ and the other moves in the direction of $S 20^{\circ} E$ with a speed of $12 \text{ kmph}$. Then the distance between the ships at the end of $2 \text{ hours}$ is (in km).

$A$ line $l$ meets the circle $x^2+y^2=61$ at points $A$ and $B$. Given that $P(-5, 6)$ is a point such that $PA=PB=10$,find the equation of line $l$.

$A$ square is inscribed in the circle $x^2 + y^2 - 2x + 4y + 3 = 0$,whose sides are parallel to the coordinate axes. One vertex of the square is

In a triangle,two vertices are $(2, 3)$ and $(4, 0)$,and its circumcentre is $(2, z)$ for some real number $z$. The circumradius is

If the circle $x^2 + y^2 - 6x - 8y + (25 - a^2) = 0$ touches the $x$-axis,then $a$ equals

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the midpoint of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$,then the radius of the circle $C$ is