Problem 53 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Polar Equation of a Conic

The polar equation of a conic with focus at the origin is given by \( r = \frac{ed}{1+e\cos\theta} \) if the directrix is a vertical line and \( r = \frac{ed}{1+e\sin\theta} \) if the directrix is a horizontal line.

Step 2: Identify the Type of Directrix

The given directrix is \(x = -5\), which is a vertical line. Therefore, the form of our equation will be \( r = \frac{ed}{1+e\cos\theta} \).

Step 3: Substitute Given Values into the Equation

Substitute the given eccentricity \(e=\frac{3}{4}\) and the distance to the directrix \(d=|-5|=5\) into the equation, obtaining \( r = \frac{\left(\frac{3}{4}\right)(5)}{1+\left(\frac{3}{4}\right)\cos\theta} \).

Step 4: Simplify the Equation

Calculate the numerator: \(\left(\frac{3}{4}\right)(5) = \frac{15}{4}\). So the equation becomes \( r = \frac{\frac{15}{4}}{1+\frac{3}{4}\cos\theta} \).

Step 5: Further Simplify the Fraction

Combine the terms: \( r = \frac{15}{4 + 3\cos\theta} \). This is the simplified polar equation of the conic.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. These curves have fascinating properties and appear in various mathematical and real-world applications. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas.

• A **circle** is a special case of an ellipse where the eccentricity is zero.
• An **ellipse** is formed when the plane cuts through both halves of the cone at a steep angle, resulting in an oval shape.
• A **parabola** is the result when the plane is parallel to the slant of the cone.
• A **hyperbola** emerges when the plane cuts through both halves of the cone at a shallow angle, giving two separate curves.

Understanding these sections helps comprehend various physical phenomena, such as planetary orbits and satellite paths. These conics can be described in both Cartesian and polar coordinates, offering flexibility depending on the problem at hand.

Eccentricity

Eccentricity is a non-negative number that describes the shape of a conic section. It determines how much the conic section deviates from being circular.
An eccentricity of:

• **0** indicates a perfect circle, the simplest conic section.
• **Less than 1** leads to an ellipse, suggesting a more stretched shape.
• **Equal to 1** corresponds to a parabola, depicting a curve that extends indefinitely in one direction.
• **Greater than 1** results in a hyperbola, indicating two separate, symmetric curves.

For the specific problem, the eccentricity given is \(e = \frac{3}{4}\), highlighting that the conic is an ellipse because the eccentricity is less than 1. The eccentricity controls the curvature of the ellipse, with smaller values leading to a more circular shape.

Directrix

A directrix is a fixed line associated with a conic section that works along with the focus to define the curve. The distance from any point on the conic to the focus is proportional to the perpendicular distance from the point to the directrix.
When dealing with polar equations, whether the directrix is a vertical or horizontal line influences the form of the equation.
In the exercise, the directrix is given as \(x = -5\), which is a vertical line. This detail guides the formula selection and is crucial in determining the positioning and form of the resulting conic section in polar coordinates. By working with the directrix, one can efficiently identify and illustrate complex curves.

Polar Coordinates

Polar coordinates offer a different way of representing points on a plane using a radius and an angle. When working with polar equations, problems often involve shapes like conic sections that are easier to describe using these coordinates rather than Cartesian coordinates.

• The coordinate \(r\) indicates the distance from the origin (pole) to the point.
• The angle \(\theta\) is measured from the positive x-axis to the line connecting the origin to the point.

This system is particularly useful for expressing curves like circles and ellipses. For conic sections with a focus at the origin, the polar equation is usually structured in the form \( r = \frac{ed}{1 + e\cos\theta} \) for vertical directrices, as shown in the given exercise. Using polar coordinates makes it more intuitive to visualize and solve problems involving round and rotationally symmetric shapes.