Finding the asymptotes of the polar curve $ r=\frac{ 5\cos^2 \theta+3 }{ 5\cos^2 \theta-1 }$

Question

Find all asymptotes ( their equations in polar or cartesian form) for the polar curve

$$ r(\theta)=\dfrac{ 5\cos^2 \theta+3 }{ 5\cos^2 \theta-1 }$$

When denominator goes to zero we have one set, but what is the other set?

Answer 1

There are four asymptotes. After converting to cartesian form they can be factored and expressed as:

$$ \pm 2 x +y = \pm \sqrt{5} $$

There are two parallel asymptote pairs.

Answer 2

You are right, values are given by

$$5\cos^2 \theta-1=0 \implies \cos \theta = \pm\frac1{\sqrt 5}\implies \theta_{1,2}=\pm \arccos \left(\frac1{\sqrt 5}\right),\: \theta_{3,4}=\pm \arccos \left(-\frac1{\sqrt 5}\right)$$

To find the asymptotes equations let consider for any direction $\theta_0$ (wlog assume $ \theta_0=\theta_{1}$)

$$y=\tan \theta_0 \, x+q \iff q=r\cos \theta (\tan \theta-\tan \theta_0)$$

then

$$q=\lim_{\theta\to \theta_0} (5\cos^2 \theta+3)\cos\theta \frac{\tan \theta-\tan \theta_0}{5\cos^2 \theta-1}=\sqrt 5$$

indeed

$$ (5\cos^2 \theta+3)\cos(\theta)\to \frac4{\sqrt 5}$$

and by l’Hopital

$$q=\lim_{\theta\to \theta_0}\frac{\tan \theta-\tan \theta_0}{5\cos^2 \theta-1}= \lim_{\theta\to \theta_0}-\frac{1}{10\cos^3\theta\sin \theta}=\frac 54$$