Problem 53 - College Algebra

Question

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-5 ; e=\frac{3}{4}\)

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{15}{4 + 3\cos(\theta)}\).

Step 1: Identify the Type of Conic Section

Since the eccentricity, \(e = \frac{3}{4}\), is less than 1, the conic with this eccentricity is an ellipse.

Step 2: Write the General Polar Equation of a Conic

The general polar equation for a conic with the focus at the origin is given by:\[r = \frac{ed}{1 + e \cos(\theta)}\]where \(d\) is the distance to the directrix, and \(e\) is the eccentricity.

Step 3: Identify the Parameters

Since the directrix is \(x = -5\), we consider that distance \(d = 5\) because the directrix distance is positive. The eccentricity \(e\) is given as \(\frac{3}{4}\).

Step 4: Substitute the Parameters

Substitute \(d = 5\) and \(e = \frac{3}{4}\) into the general polar equation:\[r = \frac{\frac{3}{4} \times 5}{1 + \frac{3}{4} \cos(\theta)}\]This then simplifies to:\[r = \frac{\frac{15}{4}}{1 + \frac{3}{4} \cos(\theta)}\]

Step 5: Simplify the Equation

Multiply the numerator and the denominator by 4 to eliminate the fractions:\[r = \frac{15}{4 + 3\cos(\theta)}\]

Key Concepts

Ellipse

An ellipse is a special type of conic section that appears as a flattened circle. It's an elongated shape with two focal points, known as foci. Every point on an ellipse is such that the sum of the distances from these two foci is constant. This property makes ellipses unique and is often used in their mathematical definition.Ellipses are encountered in everyday life in various situations, such as in the orbits of planets and the design of certain architectural structures. In the context of conic sections, the ellipse is defined by an eccentricity (\(e\)) that is less than 1. This is an important feature because it helps distinguish the ellipse from other conic sections like hyperbolas and parabolas, which have different ranges of eccentricity.

• A circle is actually a special case of an ellipse where both foci coincide at the center, and thus, the eccentricity is equal to zero.
• The length of the major axis over the minor axis determines how stretched an ellipse appears.

Eccentricity

Eccentricity is a key parameter in determining the nature of a conic section. It essentially measures how "stretched" an ellipse, or any other conic, is. The symbol for eccentricity is given as \(e\). An ellipse, by definition, has an eccentricity between 0 and 1.

• If \(e = 0\), the shape is a perfect circle, showing there is no elongation.
• If \(0 < e < 1\), the conic is an ellipse, illustrating some elongation.
• If \(e = 1\), the conic is a parabola, with even more elongation.
• If \(e > 1\), the conic is a hyperbola, and its branches are open and extend indefinitely.

In the exercise given, the eccentricity \( e = \frac{3}{4} \) reflects that the conic is an ellipse. Understanding eccentricity helps in visualizing the conic and its properties regarding symmetry and shape.

Directrix

A directrix of a conic is a line used to help define and construct the conic section. For ellipses, the directrix is typically used in conjunction with the eccentricity and focus to derive the polar equation of the conic. Although not commonly visible in plotting an ellipse on a graph, the directrix plays a vital role in maintaining the geometrical balance of the shape.In the polar equation context, the directrix is essential for determining where the conic "points," as it provides a guide against which distances are measured to value the equality of proportions relative to the eccentricity:\[r = \frac{ed}{1 + e \cos(\theta)}\]Here, \(d\) represents the positive distance to the directrix. In the exercise, you might note the directrix is given as \(x = -5\). However, the actual distance used in the equation is positive, which aligns with the measurement convention. This distance, combined with the eccentricity, helps depict the conic as an ellipse.

Polar Coordinates

Polar coordinates offer a unique way to represent points on a plane. Unlike the Cartesian system, which uses (\(x\), \(y\)) to locate a point, polar coordinates describe a location by its distance \(r\) from the origin and an angle \(\theta\) from the positive x-axis.This coordinate system is particularly useful for describing curves like ellipses, specifically when focuses are concerned. By using polar coordinates, equations of conics become more manageable, especially when the focus is central, which is why they are favored for conic sections.

• In polar form, the radial distance \(r\) can vary while \(\theta\) determines the direction.
• This system simplifies equations such as the polar equation of a conic section: \[r = \frac{ed}{1 + e \cos(\theta)}\]

Polar equations harness the circle-like nature of angular measurements and distances in a plane to represent complex shapes like ellipses, enhancing our comprehension and ability to solve related geometric problems.