Find the asymptote of rho cos phi=a cos2 phi

Find the asymptote of $\rho\cos\varphi=a\cos2\varphi$.

Step 1: Cartesian form

Since $x=\rho\cos\varphi$, the equation gives $x=a\cos2\varphi$ directly. For $y$, write $\rho=a\cos2\varphi/\cos\varphi$ and use $y=\rho\sin\varphi$: \[x=a\cos2\varphi, \qquad y=a\cos2\varphi\,\tan\varphi. \tag{1}\]

Step 2: Where the curve goes to infinity

From the second equation, $y\to\infty$ when $\tan\varphi\to\infty$ and $\cos2\varphi\neq0$, i.e.\ at $\varphi=\tfrac\pi2+n\pi$. At $\varphi=\tfrac\pi2$: $\cos2\varphi=\cos\pi=-1\neq0$, so indeed $\rho\to\infty$ and $y\to\infty$.

Step 3: Limiting abscissa

\[ x=a\cos2\varphi \xrightarrow[\varphi\to\pi/2]{} a\cos\pi=-a. \tag{2} \]

The same value $x\to-a$ is obtained at every $\varphi=\tfrac\pi2+n\pi$, since $\cos2\varphi=-1$ at each of them.

Step 4: Verification

\[a=1,\ \varphi=\tfrac\pi2-0.001\text{: }x\approx-0.999998\to-1,\ y\approx1000.0\to+\infty\ \checkmark\]