3.22. Determine that the equation $\rho=36 /(4-5 \cos \varphi)$ defines a hyperbola, and find its semi-axes.
Solution. Divide both the numerator and the denominator by 4. Transform the given equation to the form

$$
\rho=\frac{9}{\left(1-\frac{5}{4} \cos \varphi\right)}
$$

Since $\varepsilon=c / a=5 / 4>1$, the found equation defines the right branch of a hyperbola. Using formulas (3.50), $b^{2}=c^{2}-a^{2}$, and the equation defining
the eccentricity, we obtain a system of equations for finding $a$ and $b$. We have

$$
\begin{aligned}
& \frac{c^{2}}{a^{2}}=\frac{25}{16}, \quad p=\frac{b^{2}}{a}=\frac{c^{2}-a^{2}}{a}=9, a=16 \\
& \varepsilon=\frac{5}{4}, \quad c=20, b=\sqrt{c^{2}-a^{2}}=\sqrt{400-256}=12
\end{aligned}
$$