Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Ellipse} &(20,0),(4, \pi)\end{array}$$

Step-by-Step Solution

Answer

The polar equation of the conic with its focus at the pole is \(r=\frac{(5/3)24}{1+(5/3) \cos(\theta)}\).

Step 1: Compute the eccentricity and distance

From the given vertices \(20, 0\) and \(4, \pi\), we can infer that there are 2 vertices present. One vertex is on the right and the other is on the left. Thus, the ellipse should be centered at the pole. We calculate the values of \(e\) and \(p\) using the formula \(e = \frac{r_{max}}{r_{max} + r_{min}}\) and \(p = r_{max} + r_{min}\), where \(r_{max}\) and \(r_{min}\) are the maximum and minimum distances of the two vertices from the pole. Hence, \(e = \frac{20}{20+4} = \frac{5}{3}\) and \(p = 20+4 = 24\).

Step 2: Write the polar equation

Substitute the values of \(e\) and \(p\) into the general equation for an ellipse in polar coordinates that has its focus at the pole. Use the form which uses negative cosine function because the left side is the smaller side which corresponds to vertices on the \(\pi\) direction. The final polar equation for the ellipse is \(r=\frac{(5/3)24}{1+(5/3) \cos(\theta)}\). This equation represents an ellipse with a focus at the pole, a semi-major axis of \(24\) units, and an eccentricity of \(5/3\).

Key Concepts

Ellipse in Polar Coordinates

An ellipse can be described elegantly using polar coordinates, especially when one of its foci is at the origin or the pole. This is particularly useful because it simplifies the description of the ellipse's shape and position. The general form of the polar equation for an ellipse, when the focus is at the pole, is given by:\[r = \frac{ep}{1 + e \cos(\theta)}\]where:

• \(r\) is the radius vector from the pole to a point on the ellipse.
• \(e\) is the eccentricity, which indicates how much the ellipse deviates from being circular.
• \(p\) is a constant related to the semi-major axis.

In the given problem, by substituting the calculated values of \(e\) and \(p\), we arrive at the specific equation representing the ellipse. Polar coordinates help in visualizing and plotting ellipses, which might be awkward to handle using Cartesian coordinates.

Eccentricity

Eccentricity is a crucial parameter in conic sections, including ellipses. It quantifies the degree to which an ellipse deviates from being a perfect circle:

• If the eccentricity \(e = 0\), the shape is a circle.
• For \(0 < e < 1\), the conic is an ellipse.

In our exercise, the eccentricity is calculated using the formula \(e = \frac{r_{\text{max}}}{r_{\text{max}} + r_{\text{min}}}\), where \(r_{\text{max}}\) and \(r_{\text{min}}\) are the larger and smaller of the distances from the pole to the ellipse's vertices.Here, the computed eccentricity of \(\frac{5}{3}\) indicates a prominent elongation, highlighting that it is an ellipse and not a circle. Understanding eccentricity helps to assess the "flattening" of the ellipse and plays a key role in its mathematical representation.

Focus at the Pole

When a conic section, like an ellipse, has its focus at the pole in polar coordinates, it changes how we approach its equation and visualization. In these scenarios, the conic is created such that the focus is at the origin point in polar terms:

• This setup keeps the calculations centered around the pole, simplifying aspects like symmetry.
• Focus at the pole means one vertex has maximum radius, while the opposite vertex has minimum radius.

This uniquely positions the ellipse in the polar grid, allowing for straightforward calculations. In our exercise, calculating the eccentricity and equating it within this framework gave us a simplified yet descriptive version of the ellipse's equation.

Semi-Major Axis

The semi-major axis is one of the essential elements in defining the size and shape of an ellipse. It is essentially half of the longest diameter of the ellipse. In polar equations, though, describing the semi-major axis involves a relationship with the parameter \(p\), especially relevant when the focus is at the pole. In our executive example:

• \(p = 24\), as calculated, represents twice the semi-major axis's length in polar coordinates.
• The longest span from one side of the ellipse to the other is controlled by this calculation.

The semi-major axis thus defines how "stretched" out the ellipse is, and in conjunction with other parameters like eccentricity, it tells a full story about the ellipse's geometry and orientation. Understanding the semi-major axis is vital for grasping how large the ellipse is and how it will appear on a polar grid.