Line in polar coordinates

Question

I just wanted to clarify something. A line in polar coordinates has the parameterization of $\theta = k\pi$ for $k \in \mathbb{R}$ right? Or am I missing something?

Answer 1

The line described by your polar equation goes through the origin and is at a constant angle from the $x$ axis.

Now for a line perpendicular to a radial line making a constant angle $\phi$ with the $x$ axis, at a distance $r_0\neq 0$ from the origin, the general polar equation is

$$r(\theta)={r_0\over \cos(\theta-\phi)}$$

Answer 2

Hint: Let $x = r \cos \theta$ and $y= r \sin \theta$ and see if you can find the polar equations for

$$y = mx + b \\ y = x + b \\ y = x$$ For the third equation, what do you notice?

Answer 3

$\theta=k\pi,k\in\mathbb R$ is just $\theta\in\mathbb R$.

This describes the locus of points at any distance from the origin, in the direction of $\theta$. Hence it is a half-line from the origin, possibly not what you want.

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A line of Cartesian equation

$$ax+by+c=0$$ becomes

$$ar\cos\theta+br\sin\theta+c=0$$ or

$$r=-\frac c{a\cos\theta+b\sin\theta}.$$

Answer 4

If follows from polar coordinate definition

$$ \theta = constant = c$$

is a line in polar coordinates through the origin. In particular

$$c= 0, k\pi,\quad k \in \mathbb{R}$$ are lines along x-axis and $$c= 1, \pi(2k-1) $$

are lines along y-axis.