Problem 10 - Precalculus Mathematics 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 \( r = \frac{1.2}{1 - 0.4 \cos \theta} \).

Step 1: Understand the properties of the ellipse

An ellipse in polar form is expressed with its focus at the origin and its general equation is \( r = \frac{ed}{1 - e\cos \theta} \), where \( e \) is the eccentricity, and \( d \) represents the distance from the focus to the directrix. In an ellipse, the eccentricity \( e < 1 \).

Step 2: Identify given values

From the problem, we are given the eccentricity \( e = 0.4 \). We also know the polar coordinates of the vertex are given as \( (2,0) \), where the radius \( r = 2 \) when \( \theta = 0 \).

Step 3: Establish the relationship at the vertex

At the vertex \( (2,0) \), use the equation \( r = \frac{ed}{1 - e\cos \theta} \) and substitute the known values: \( r = 2 \), \( e = 0.4 \), \( \theta = 0 \). This simplifies to \( 2 = \frac{0.4d}{1 - 0.4 \cdot 1} \).

Step 4: Solve for the directrix distance \( d \)

Simplify the equation from Step 3: \( 2 = \frac{0.4d}{0.6} \). Multiply both sides by 0.6 to get \( 1.2 = 0.4d \). Solve for \( d \) by dividing both sides by 0.4, yielding \( d = 3 \).

Step 5: Write the polar equation of the ellipse

With \( e = 0.4 \) and \( d = 3 \), substitute back into the polar form \( r = \frac{ed}{1 - e\cos \theta} \). This gives \( r = \frac{0.4 \times 3}{1 - 0.4 \cos \theta} = \frac{1.2}{1 - 0.4 \cos \theta} \). The equation \( r = \frac{1.2}{1 - 0.4 \cos \theta} \) is the polar equation of the ellipse with the given conditions.

Key Concepts

Conic Sections

Conic sections are fascinating curves resulting from the intersection of a plane with a double-napped cone. These curves can be circles, ellipses, parabolas, or hyperbolas, each with distinct properties. The shape of the conic depends on the angle of the cut and the distance from the focus to the directrix. Understanding conic sections is integral when exploring various applications in physics, engineering, and astronomy. Here are some key aspects:

• They are defined by a focus, directrix, and eccentricity.
• The type of conic section (circle, ellipse, etc.) is determined by its eccentricity (\(e\)).
• Eccentricity is crucial in defining the shape; for ellipses, \(0 < e < 1\).

These properties create a rich framework for analyzing polar equations, like in the case of an ellipse with a specific vertex and eccentricity.

Eccentricity

Eccentricity is a crucial parameter in defining the shape of a conic section. The value of eccentricity determines whether the conic section is an ellipse, a parabola, or a hyperbola. In particular:

• Ellipses have eccentricity values less than 1 (\(0 < e < 1\)).
• When \(e = 1\), the conic is a parabola.
• For hyperbolas, the eccentricity exceeds 1 (\(e > 1\)).

The exercise provides an eccentricity of \(0.4\), indicating the conic section is an ellipse. Understanding this helps describe how stretched the ellipse is, with greater values of \(e\) indicating a more elongated shape. This parameter influences how the curve responds to changes in polar coordinates and is pivotal in solving the sample exercise.

Ellipse Equations

Ellipse equations in polar format are essential for understanding the relationship between the geometric properties and algebraic expressions. The general form used in the exercise is \( r = \frac{ed}{1 - e\cos \theta} \), where:

• \(r\) represents the radius, or distance from the focus to a point on the ellipse.
• \(e\) is the eccentricity that affects the ellipse's shape.
• \(d\) is the distance from the focus to the directrix.
• \(\theta\) is the angle in polar coordinates.

In the solution, substituting known values \(e = 0.4\), vertex \((2,0)\), and using relationships at specific points aids in deriving the polar equation \( r = \frac{1.2}{1 - 0.4 \cos \theta} \). The formula intricately links the geometry of the ellipse with algebraic manipulation to solve real-world problems where such paths occur.