Problem 3 - 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 \(\frac{2}{3},\) directrix \(x=3\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{6}{3 - 2\cos\theta} \).

Step 1: Understand the Equation Format for Conics in Polar Coordinates

For a conic section with its focus at the origin, the polar equation can be written as \( r = \frac{ed}{1 \pm e\cos\theta} \) or \( r = \frac{ed}{1 \pm e\sin\theta} \), where \( e \) is the eccentricity and \( d \) is the distance of the directrix from the origin.

Step 2: Identify Parameters for the Given Conditions

The problem states that the conic is an ellipse with eccentricity \( e = \frac{2}{3} \). The directrix is given as \( x = 3 \), which means the distance \( d = 3 \). Since it's along the x-axis, we will use \( \cos \theta \).

Step 3: Substitute Values into the Polar Equation Format

Substitute the given values into the polar form of the conic section: \( r = \frac{3 \times \frac{2}{3}}{1 - \frac{2}{3} \cos\theta} \).

Step 4: Simplify the Polar Equation

Calculate the numerator: \( 3 \times \frac{2}{3} = 2 \). Thus, the equation becomes \( r = \frac{2}{1 - \frac{2}{3} \cos\theta} \). Further simplify the denominator to write the equation as \( r = \frac{2}{1 - \frac{2}{3} \cos\theta} = \frac{2}{\frac{3-2\cos\theta}{3}} = \frac{6}{3-2\cos\theta} \).

Key Concepts

Conic Sections

Conic sections are curves that you get by slicing a cone with a plane. Depending on the angle and position of the slice, you can get one of four shapes: ellipse, parabola, hyperbola, or circle. In polar coordinates, these shapes can be represented using specific equations.
The main formula used is often in the form:

• Ellipse: \[ r = \frac{ed}{1 \pm e\cos\theta} \text{ or } r = \frac{ed}{1 \pm e\sin\theta} \]
• Parabola, hyperbola and other conics follow similar formats, changing mainly with the signs and values of constant components.

For our ellipse, we are particularly interested in how the elements such as eccentricity and directrix influence the shape in the polar coordinate system.

Eccentricity

Eccentricity measures how circular or stretched out a conic section is. For ellipses, the eccentricity (\( e \) ) is always between 0 and 1.
Here are the key points:

• When \( e = 0 \), the shape is a perfect circle.
• As \( e \) increases towards 1, the ellipse becomes more elongated.

In our exercise, the ellipse has an eccentricity of \( \frac{2}{3} \), meaning it's not too stretched out and relatively closer to being circular. This affects how the equation is formed and how the ellipse appears in polar coordinates.

Directrix

The directrix is a fixed line used in the definition of a conic section. It helps determine how "wide" or "narrow" the conic is and interacts with the focus, another key element. For polar equations, the distance from the origin to the directrix is crucial.
In our exercise:

• The directrix is given as \( x = 3 \) , meaning it's located 3 units from the origin along the x-axis.
• This situates the ellipse relative to the origin when plotting in polar coordinates.

By substituting this distance into our equation \[ r = \frac{ed}{1 \pm e\cos\theta} \] , we influence the curve's specific positioning and orientation in the plane.