Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Ellipse} &\left(2, \frac{\pi}{2}\right),\left(4, \frac{3 \pi}{2}\right)\end{array}$$

Step-by-Step Solution

Answer

The polar equation of the given conic with its focus at the pole is \(r = 0\).

Step 1: Identify the characteristics of the ellipse

One can identify that this is an ellipse with a vertical axis of symmetry since the two vertices are mirrored along the x-axis. The foci of ecliidic sections are always inside the conic section, and the fact that one is at the origin provides an additional fact about the given ellipse. Since foci in an ellipse are always symmetrically distributed along the major axis, we know that the center of this ellipse is at the origin, and its major axis is the line \(y=\pi/2\) and \(y=3\pi/2\).

Step 2: Calculate the lengths of semi-axes

Knowing that an eccentricity of an ellipse is less than 1, we can establish the following relationship between the distances from a focus to the ellipse and its center \(e = \frac{c}{a}\). That means the distance between a focus and the center of the ellipse is \(a*e = c\). In this case, \(a=2\) (minor semi-axis), \(c=2\) (distance between a focus and the center), so \(e=1\) (eccentricity of an ellipse). Therefore, by substituting \(e = 1\) and \(c = 2\) into the eccentricity relationship, we can solve for \(a\) (major semi-axis), \(a = \frac{c}{e} = \frac{2}{1} = 2\). Hence our major semi-axis is 2 and our minor semi-axis is 2.

Step 3: Write down the polar equation

Since we have determined that their semi-major and semi-minor axes are equal, the given conic section is a circle. Therefore, the polar equation for this figure is \(r = a(1-e) = 2(1-1) = 0\).

Key Concepts

Ellipse

An ellipse is one type of conic section that is shaped like an elongated circle. It is defined mathematically by a set of points in a plane, such that the sum of the distances to two fixed points, called foci, is constant. This balancing act between distances means the shape bulges outward to form a recognizable oval curve.

In polar coordinates, an ellipse can be related to its foci, one of which is often placed at the origin or 'pole'. The equation that governs an ellipse in polar form is generally given by:

• \( r = \frac{a(1-e^2)}{1 + e\cos(\theta)} \)

where \( e \) is the eccentricity and \( a \) is the semi-major axis. In the particular problem at hand, we are dealing with an ellipse with its semi-major and semi-minor axes equal, revealing it to be a special case: a circle.

Focus at the Pole

When talking about conic sections like ellipses, one often chooses to place the focus at the pole, or the origin point of a polar coordinate system. This makes the math a bit simpler since the focus serves as a central reference for calculating distances and plotting points.

By placing the focus at the pole, it can be easier to understand some central characteristics of the conic. It keeps the conditions neatly aligned to specific polar equations, helping streamline the calculation of important elements like eccentricity and the lengths of axes.

In our example, positioning the focus at the pole gives us direct insights into central symmetries and alignments, notably along the vertical axis of symmetry, leading us to pinpoint that our conic section turns into a perfect circle.

Eccentricity

Eccentricity in conic sections defines just how elongated or circular a shape actually is. It can be likened to a measure of 'ovalness'. For an ellipse, the eccentricity is always between 0 and 1. The closer it is to 0, the more circular the ellipse is; the closer it is to 1, the more stretched out it becomes.

This relationship can be described as:

• \( e = \frac{c}{a} \)

where \( c \) is the distance from the center to a focus, and \( a \) is the length of the semi-major axis. In the given problem, the ellipse surprisingly has an eccentricity, \( e \), equal to 1, contrary to a typical ellipse. This outcome signifies a circle, indicating unique equal semi-major and minor axes.

Major and Minor Axes

The major and minor axes are crucial for understanding the proportions and geometry of an ellipse.

• The major axis is the longest diameter of the ellipse, running through both foci.
• The minor axis is the shorter diameter, perpendicular to the major axis at the ellipse's center.

In our exercise, we observe both the major and minor axes having an equal length, confirming the transformation of the shape from an ellipse to a circle.

Given that their lengths are the same, this conic section behaves as a circle with a consistent diameter. Such characteristics help in deriving the polar equation directly: \( r = a(1-e) \), which simplifies to \( r = 0 \) in our situation, aligning with the major and minor axis characteristic of a circle.