Problem 50 - Precalculus with Limits

Question

In Exercises 39-54, find a polar equation of the conic with its focus at the pole. \(\textit{Conic}\) Ellipse \(\textit{Vertex or Vertices}\) \((2, \pi/2), (4, 3\pi/2)\)

Step-by-Step Solution

Answer

The polar equation of the given ellipse with its focus at the pole and vertices at \((2, \pi/2)\) and \((4, 3\pi/2)\) is \(r = 2\).

Step 1: Identify the vertex and calculate the semi-major axis length (\(a\))

From the given vertices \((2, \pi/2)\) and \((4, 3\pi/2)\), we take the radial distance of any vertex as the value of the semi-major axis \(a\). Therefore, \(a = 2\).

Step 2: Calculate the semi-minor axis length (\(b\))

The semi-minor axis length is the distance from the center to the other vertex, which gives us \(b = 4 - a = 4 - 2 = 2\).

Step 3: Develop the equation of the ellipse

Using the polar form of the equation of an ellipse centered at the pole, which is \(r = \frac{a(1 - e^2)}{1 - ecos(\Theta)}\), given that \(e = \sqrt{1 - (b/a)^2}\) is the eccentricity, and we have calculated \(a\) and \(b\), we substitute these values into the equation to get the equation of the ellipse. \n Therefore, \(e = \sqrt{1 - (2/2)^2} = 0\) and the polar equation of the ellipse is \(r = \frac{2(1 - 0^2)}{1 - 0cos(\Theta)} = 2\).

Key Concepts

Ellipse

An ellipse is a geometric shape that resembles a squashed circle. It is characterized by having two focal points, referred to as foci. When looking at an ellipse, each point on the shape ensures that the sum of the distances to the two foci is constant. This unique property is what sets an ellipse apart from a simple circle, where each point is equidistant from a single center point.
In polar coordinates, ellipses can be defined by their equation. Unlike the Euclidean plane where you have to deal with x and y coordinates, polar coordinates deal with angles and radial distances, making it an interesting way to represent curves like ellipses.
This can make polar forms particularly useful in fields such as astronomy, where celestial movements often follow conic sections like ellipses. Understanding how to convert and work with these equations in polar form is vital if you're looking to delve deeper into topics like orbital mechanics or even sophisticated graphics programming.

Polar Coordinates

Polar coordinates provide a way to describe the location of a point on a plane using an angle and a distance. Instead of the typical x and y coordinates, a point is identified by how far it is from a central point (called the pole) and the angle it makes with a reference direction, usually designated as the positive x-axis.
This system is particularly effective for dealing with curves that wrap around each other or rotate around a center. Converting from rectangular to polar coordinates can simplify problems that have symmetry around a point, such as those involving circles and ellipses like we do here.
In the exercise you encountered, utilizing polar coordinates allows us to express a conic section's equation in terms that easily account for its orientation and stretching. This makes it easier to develop and understand the nature of curves whose orientation doesn't align perfectly with the x and y axes.

Eccentricity

Eccentricity is a key concept that helps define the shape and type of a conic section, such as the ellipse, circle, hyperbola, or parabola. The eccentricity of a conic section is a non-negative number that measures how much the conic section deviates from being circular.
For an ellipse, the eccentricity is a value between 0 and 1. A circle is essentially a special type of ellipse with an eccentricity of 0, meaning it has no deviation and is perfectly round. The further the eccentricity is from 0, the more stretched or elongated the ellipse becomes.
In the exercise, the calculation showed an eccentricity of 0, indicating that the conic section in question is a circle. Despite being labeled as an ellipse in the problem, it shows that all points are equidistant from the pole, just like a circle. Understanding how eccentricity affects shape can deepen your understanding of geometric curves and their applications in both pure mathematics and applied sciences, such as optics and orbital studies.