11. Sides $A B$ and $C D$ of quadrilateral $A B C D$ are perpendicular and are diameters of two equal tangent circles with radius $r$. Find the area of quadrilateral $A B C D$, if $\frac{|B C|}{|A D|}=k$.
11. Let $M$ be the point of intersection of the lines $A B$ and $C D$, and let $O_{1}$ and $O_{2}$ be the centers of the given circles. Suppose $|B M| = |x|$, $|M C| = |y|$, and choose the signs of $x$ and $y$ such that the following relations hold:

$$
\begin{aligned}
|B C|^{2}= & x^{2}+y^{2}, \quad|A D|^{2}=(x+2 r)^{2}+(y+2 r)^{2} \\
& \left|O_{1} O_{2}\right|^{2}=(x+r)^{2}+(y+r)^{2}
\end{aligned}
$$

We obtain the system

$$
\begin{gathered}
\left\{\begin{array}{l}
x^{2}+y^{2}=k^{2}(x+2 r)^{2}+k^{2}(y+2 r)^{2}, \\
4 r^{2}=(x+r)^{2}+(y+r)^{2},
\end{array}\right. \\
\text { from which } x+y=r \frac{1-5 k^{2}}{1+k^{2}}, \text { therefore } \\
S_{A B C D}=\left|\frac{(x+2 r)(y+2 r)-x y}{2}\right|=\left|r(x+y)+2 r^{2}\right|=3 r^{2}\left|\frac{1-k^{2}}{1+k^{2}}\right| \cdot \\
\text { Answer: } S=3 r^{2}\left|\frac{1-k^{2}}{1+k^{2}}\right| .
\end{gathered}
$$