Question

For the following exercises, sketch the graph of each conic. $$r=\frac{1}{1-\cos \theta }$$

Short answer

The graph is a parabola opening to the right with the focus at the pole and directrix at $r=1$.

Step-by-step solution

Identify the Conic

First, we need to identify what kind of conic section is represented by the given polar equation. The general form of a polar equation for a conic is $r=\frac{ed}{1-e\cos \theta }$. Comparing it to the equation $r=\frac{1}{1-\cos \theta }$, we see that $e=1$ and $d=1$. Thus, the conic is a parabola with the focus at the pole.

Recognize the Directrix

With the conic confirmed as a parabola, the directrix will be a line parallel to the polar axis and located at a distance $d$ from the pole. Since $d=1$, the directrix is the vertical line $r=1$ in the coordinate system.

Sketch the Graph

To sketch the graph, observe that for every $\theta$, the distance from the focus (pole) to the curve is $r=\frac{1}{1-\cos \theta }$, which can be rewritten as $r=\frac{1}{2{\sin }^2(\theta /2)}$ using trigonometric identities. This confirms all points $(r,\theta )$ are equidistant from the focus and directrix. Start sketching from the point where $\theta =0$, and point (1,0) converges into the x-axis. The graph will open to the right, as $\theta$ increases in both directions, indicating all possible points where $\cos \theta eq1$.

Consider Special Positions

Think about where $\theta =\frac{\pi }{2}$ and $\theta =-\frac{\pi }{2}$. When $\theta =\pi$, $\cos \theta =-1$ and points get infinitely far away, which aligns with the vertex of the parabola. The main body of the parabola thus runs along the positive side of the polar axis (x-axis in Cartesian coordinates).

Key concepts

Understanding Parabolas in Polar Conic Sections

A parabola is a type of conic section that you can think of as the set of all points equidistant from a specific point called the focus, and a line called the directrix. In polar coordinates, the general equation for a parabola is represented as $r=\frac{ed}{1-e\cos \theta }$. In this equation, $e$ is the eccentricity, which is equal to 1 for parabolas, and $d$ represents the distance from the focus to the directrix.

In the given exercise, the equation $r=\frac{1}{1-\cos \theta }$ confirms it is indeed a parabola. Here, both $e$ and $d$ are equal to 1, which tells us that the conic is a parabola with its focus at the origin (the pole) and the directrix as the line $r=1$. This setup causes the parabola to open to the right, extending along the positive x-axis as $\theta$ increases in polar coordinates. The parabola's interesting properties, such as having a symmetrical shape and infinite arms, arise from these geometric relationships.

Polar Coordinates and Their Application to Conics

Polar coordinates provide a unique way of describing the position of points in a plane using a distance from a reference point and an angle from a reference direction. This differs from Cartesian coordinates that simply use $x$ and $y$ coordinates. In the polar system, a point is represented by $(r,\theta )$, where $r$ is the radius, or the distance from the pole, and $\theta$ is the angle measured from the positive x-axis.

This system is quite useful when graphed curves like conics, such as parabolas, circles, ellipses, and hyperbolas, because it can often simplify the equations needed to represent them. In our example, the polar equation $r=\frac{1}{1-\cos \theta }$ describes a parabola. As $\theta$ changes, different values of $r$ are calculated, plotting points around the polar axis that ultimately form the shape of the parabola. Polar coordinates simplify the visualization of such geometric figures because they naturally align with their radial symmetry and directions.

Leveraging Trigonometric Identities

Trigonometric identities are vital tools in rewriting and simplifying equations, especially in polar coordinates. In the given exercise, the transformation from $r=\frac{1}{1-\cos \theta }$ to the form $r=\frac{1}{2{\sin }^2(\theta /2)}$ is achieved using these identities, specifically the half-angle identities.

Understanding and using identities like $\cos \theta =1-2{\sin }^2(\theta /2)$ allows us to rearrange and express the equations in ways that highlight certain properties or simplify the analysis of a conic section. For instance, in polar equations, expressing terms in sine instead of cosine may reveal symmetries or help in visualizing how a graph behaves as $\theta$ changes. These transformations are powerful, making complex geometry more approachable and enabling deeper insights into the shapes and relations of curves in polar coordinates.