12.5 Conic Sections in Polar Coordinates

Conic Sections in Polar Coordinates

Instead of describing conics with $x$ and $y$ in Cartesian form, you can use polar coordinates to represent all four conic sections with a single equation type. The polar form places one focus at the origin and ties together eccentricity, the focus, and the directrix in one clean expression. This makes it especially useful for problems involving orbits, satellite paths, and other scenarios where distance from a central point matters.

Classification of Polar Conics

The general polar equation for a conic section is:

$r = \frac{ep}{1 \pm e\cos\theta}$

There are two key parameters here:

• Eccentricity ($e$) controls the shape of the conic. Its value tells you exactly which type of curve you're dealing with.
• Focal parameter ($p$) is the perpendicular distance from the focus (at the origin) to the directrix. It controls the size of the conic.

The $\pm$ in the denominator determines which side of the origin the directrix sits on. You may also see $\sin\theta$ instead of $\cos\theta$, which rotates the conic so the directrix is perpendicular to the vertical axis rather than the horizontal one.

Classifying by eccentricity:

For example, with $e = 0.5$ and $p = 4$:

$r = \frac{(0.5)(4)}{1 - 0.5\cos\theta} = \frac{2}{1 - 0.5\cos\theta}$

Since $0 < 0.5 < 1$, this is an ellipse. If you changed $e$ to $1$, the same structure gives you a parabola: $r = \frac{4}{1 - \cos\theta}$.

Graphing Conics in Polar Coordinates

To graph a polar conic, you plot points $(r, \theta)$ where $r$ is the radial distance from the origin and $\theta$ is the angle measured counterclockwise from the positive $x$-axis (the polar axis).

Steps for graphing:

1. Choose key values of $\theta$, typically $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$, and so on through $2\pi$.
2. Plug each $\theta$ into the polar equation and calculate $r$. If $r$ comes out negative, the point is plotted in the opposite direction.
3. Plot each $(r, \theta)$ point on polar graph paper (or a polar coordinate system).
4. Connect the points with a smooth curve.

Start by finding $r$ at $\theta = 0$ and $\theta = \pi$. These give you the closest and farthest distances from the focus along the polar axis, which helps you set the scale of your graph quickly.

Symmetry shortcuts:

• If the equation uses $\cos\theta$, the conic is symmetric about the polar axis (the horizontal line $\theta = 0$). You only need to plot from $0$ to $\pi$ and reflect.
• If the equation uses $\sin\theta$, the conic is symmetric about the line $\theta = \frac{\pi}{2}$ (the vertical axis).

Focus, Directrix, and Eccentricity in Polar Form

These three elements are built directly into the polar equation, so understanding how they connect is essential.

Focus: In the polar form, one focus is always at the origin $(0, 0)$. This is what makes polar form so convenient: distances are measured directly from a focus.

Directrix: A fixed line perpendicular to the polar axis that doesn't pass through the focus. The focal parameter $p$ is the distance from the focus to this directrix. When the equation uses $1 - e\cos\theta$, the directrix is to the left of the focus at $x = -p$. When it uses $1 + e\cos\theta$, the directrix is to the right at $x = p$.

Eccentricity: Defined as the ratio of distances for any point on the conic:

$e = \frac{\text{distance from point to focus}}{\text{distance from point to directrix}}$

This ratio is constant for every point on the curve. That's actually the definition of a conic section in terms of focus and directrix. When $e < 1$, each point is closer to the focus than to the directrix (ellipse). When $e = 1$, the distances are equal (parabola). When $e > 1$, each point is farther from the focus than from the directrix (hyperbola).

Working with the Polar Form

The polar equation $r = \frac{ep}{1 \pm e\cos\theta}$ is powerful because it unifies all conic types. Here's how to use it in practice:

Identifying a conic from its equation:

1. Get the equation into the standard form $r = \frac{ep}{1 \pm e\cos\theta}$ (or with $\sin\theta$). The denominator must start with $1$, so divide numerator and denominator by any leading coefficient if needed.
2. Read off $e$ from the denominator. That's the coefficient in front of $\cos\theta$ or $\sin\theta$.
3. Find $p$ by dividing the numerator by $e$.
4. Classify using the eccentricity table above.

Example: Identify $r = \frac{6}{2 + 3\cos\theta}$.

1. Divide top and bottom by 2: $r = \frac{3}{1 + 1.5\cos\theta}$
2. So $e = 1.5$
3. The numerator is $ep = 3$, so $p = \frac{3}{1.5} = 2$
4. Since $e = 1.5 > 1$, this is a hyperbola with the directrix 2 units from the focus.