How to find asymptotes of this weird hyperbola

Question

How does one find equations for the asymptotes of hyperbolae of the form:

$$k \mu^2-2cq\mu-\sigma^2d+q^2z=0$$

where $\mu$ is the dependent variable, $\sigma$ is the independent variable, and the rest are parameters.

Answer 1

Let $x = \mu - \frac{cq}{k}$. Then the equation becomes $$kx^2 -d\sigma^2 = \frac{c^2q^2}{k} -q^2 z $$ The quantity on the right hand side is irrelevant for the purpose of finding the asymptotes, which are the lines $$ \sigma = \pm \sqrt{\frac{k}{d}} x $$
or in terms of $\mu$, $$ \sigma = \pm \sqrt{\frac{k}{d}} (\mu - \frac{cq}{k}) $$

Answer 2

Assume that there is an oblique asymptote so that $\mu$ and $\sigma$ go to infinity simultaneously. If you only keep the terms of the highest degrees,

$$k\mu^2-d\sigma^2=0$$ gives you the direction of the two asymptotes,

$$\sigma=\pm\sqrt{\frac kd}\mu.$$

Now to obtain the intercept, plug

$$\sigma=\pm\sqrt{\frac kd}\mu+p$$ and solve for $p$:

$$k \mu^2-2cq\mu-\left(\pm\sqrt{\frac kd}\mu+p\right)^2d+q^2z=0$$ only keeping the high-order terms in $\mu$, after simplification:

$$k \mu^2-2cq\mu-k\mu^2\pm2\sqrt{kd}\mu p=0$$ so that

$$p=\frac{cq}{\sqrt{kd}}.$$