$r=5 \sec(\theta)$ into rectangular

Question

I need to convert $r=5\sec(\theta)$ into rectangular form. I think I need to multiply both sides by $r$, because $r^2 = x^2 + y^2$, but i'm not sure how to convert $r=5\sec(\theta)$ in terms of $x$ and $y$. $x=rcos(\theta)$ but how do I use that? Thanks.

Answer 1

Assuming $\theta \neq n\pi + \pi/2$, note that $r = 5 \sec(\theta) \implies r \cos(\theta) = 5 \implies x = 5$.

Answer 2

In general, we have $$\begin{align*} x &= r\cos\theta \\ y &= r\sin\theta \end{align*}$$ so in this case, we have $$\begin{align*}x &= 5\sec\theta\cos\theta \\ y &= 5\sec\theta\sin\theta.\end{align*}$$

For $\theta\neq n\pi + \frac\pi2$, we can see that the $x$-equation simplifies to $x = 5$, and that the $y$-equation simplifies to $y = 5\tan\theta$. Because the range of $\tan\theta$ is $\mathbb{R}$, for any value $y^*$ of $y$, there is some value of $\theta$ such that $y(\theta) = y^*$.

Therefore, this is a vertical line at $x=5$.