Problem 8 - Precalculus: Mathematical for Calculus

Question

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.4,\) vertex at \((2,0)\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{1.68}{1 - 0.4\cos\theta} \).

Step 1: Understand the Problem

We need to write a polar equation for an ellipse with a given eccentricity and a vertex at a specific point. The ellipse has its focus at the origin.

Step 2: Recall the Polar Equation of a Conic

The general polar equation for a conic with focus at the origin is \( r = \frac{ed}{1 - e\cos\theta} \), where \( e \) is the eccentricity and \( d \) is the directrix.

Step 3: Use Eccentricity and Conic Type

Since the eccentricity \( e = 0.4 \) and is less than 1, the conic is an ellipse. For ellipses, the polar equation can also be written as \( r = \frac{a(1-e^2)}{1 - e\cos\theta} \).

Step 4: Determine the Value of \( a \)

The vertex at \((2, 0)\) indicates that the semi-major axis \( a = 2 \).

Step 5: Construct the Polar Equation

Substitute \( a = 2 \) and \( e = 0.4 \) into the polar equation for the ellipse: \( r = \frac{2(1 - 0.4^2)}{1 - 0.4\cos\theta} \).

Step 6: Simplify the Equation

Simplify the equation: \( r = \frac{2(1 - 0.16)}{1 - 0.4\cos\theta} = \frac{2 \times 0.84}{1 - 0.4\cos\theta} = \frac{1.68}{1 - 0.4\cos\theta} \).

Key Concepts

Eccentricity

Eccentricity is a key concept when it comes to understanding conic sections. It tells us how "stretched" or "squished" a conic section is. For different types of conic sections, eccentricity behaves differently. The value of eccentricity, denoted as \( e \), helps determine the shape of the conic:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), the conic section is an ellipse.
• If \( e = 1 \), it forms a parabola.
• If \( e > 1 \), it becomes a hyperbola.

As you can see, for the ellipse in this exercise, the eccentricity is \( e = 0.4 \). Since this value is between 0 and 1, it is indeed classified as an ellipse. The smaller the eccentricity value, the more circular the ellipse appears. In this case, an eccentricity of 0.4 means it's slightly stretched, but still closer to being circular than elongated.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone at different angles. These intersections give rise to the different types of conic sections which are the circle, ellipse, parabola, and hyperbola. Each type has unique characteristics defined by their specific geometric properties.
Conic sections are used in various fields such as physics, astronomy, engineering, and architecture.

• Circle: A special case of an ellipse where the eccentricity is 0.
• Ellipse: Formed when the plane cuts through the cone at an angle, not perpendicular to the base, yet not parallel to the side.
• Parabola: The plane is parallel to the cone's side.
  
• Hyperbola: Occurs when the plane cuts through both halves of the cone.

The focus and directrix help specify the conics in polar equations. In this exercise, we have focus at the origin and the type identified by the eccentricity, which tells us it's an ellipse.
Understanding these basics is crucial for mastering polar equations involving conics.

Ellipse

An ellipse is a type of conic section defined by its geometric properties and its relationship with the focus and eccentricity. In polar coordinates, an ellipse's equation takes a specific form because its focus is at the origin.
The general polar equation for an ellipse is:\[ r = \frac{a(1-e^2)}{1 - e\cos\theta}\]where:

• \( a \) is the semi-major axis (the longest diameter of the ellipse).
• \( e \) is the eccentricity.
• \( \theta \) is the angle from the polar axis.

In our example, we know the semi-major axis is 2, given by the vertex. The polar equation becomes specific to these known values and simplifies to \( r = \frac{1.68}{1 - 0.4\cos\theta} \).
Ellipses have two foci, and the sum of the distances from any point on the ellipse to these foci is constant. Understanding this concept helps solve polar equation problems involving ellipses, like the one in this exercise.
Enhancing your grasp of these properties is essential for advanced geometry and helps in dealing with real-world applications like planetary orbits and optical systems.