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 $$\$fTrueac{2}{3},$$ directrix $x=3$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{2}{1 + \frac{2}{3}\cos\theta} \).

Step 1: Identify the type of conic and its eccentricity

The conic is given as an ellipse with an eccentricity of \(\frac{2}{3}\). Recall that for an ellipse, the eccentricity \(e\) must satisfy \(0 < e < 1\). This confirms that the eccentricity \(\frac{2}{3}\) is valid for an ellipse.

Step 2: Recall the polar equation of a conic

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

Step 3: Determine the given directrix and distance

The directrix is provided as \(x = 3\). Since the directrix is vertical and given as \(x = 3\), the distance \(d\) from the focus (at the origin) to the directrix is \(3\).

Step 4: Write the polar equation using the parameters

Given the eccentricity \(e = \frac{2}{3}\) and the directrix at \(x = 3\), the polar equation is calculated as follows: Use \(d = 3\), hence the equation is \( r = \frac{(\frac{2}{3})(3)}{1 + (\frac{2}{3})\cos\theta} \).

Step 5: Simplify the polar equation

Simplifying the expression, we get \( r = \frac{2}{1 + \frac{2}{3}\cos\theta} \). This polar equation represents the ellipse with the given conditions.

Key Concepts

Understanding Ellipses

An ellipse is one of the conic sections, which also includes circles, parabolas, and hyperbolas. What makes an ellipse unique is its shape. It's like an elongated circle and has two main points called foci. The sum of the distances from any point on the ellipse to the two foci is always constant. This property distinguishes an ellipse from other conic sections.
You can envision it as a stretched loop. Notably, when you join certain points of interest on an ellipse through line segments, such as from the ellipse center to a point on the border, you create semi-major and semi-minor axes.
Here are some important features of ellipses:

• Major Axis: The longest diameter of the ellipse, running through the center and both foci.
• Minor Axis: The shortest diameter, perpendicular to the major axis.
• Foci (plural for focus): Two points inside the ellipse that help in defining its shape.

Overall, ellipses are commonly observed in astronomy, especially in the orbits of planets and moons, making them important beyond just math.

Diving into Eccentricity

Eccentricity is a measure that describes how squished or stretched a conic section is, when compared to a circle. In the case of an ellipse, the eccentricity, denoted as \(e\), determines how close the ellipse is to being a perfect circle.
For ellipses, the eccentricity always lies between 0 and 1:

• If \(e = 0\), it's a perfect circle.
• If \(0 < e < 1\), it's an ellipse.

In our exercise, the eccentricity is given as \(\frac{2}{3}\). This tells us that the ellipse is somewhat elongated but not excessively so. Eccentricity helps to plot and predict the behavior and appearance of an ellipse when examining its equation in polar coordinates.
You can think of eccentricity as a "stretch factor" that tweaks the ellipse's shape, affecting how far apart the foci are within the ellipse.

Exploring the Directrix

The directrix is an essential element in defining the shape and orientation of conic sections. It is a straight line relative to which the conic is drawn. In the context of our exercise, we're given a vertical directrix, denoted by the equation \(x = 3\).
This directrix has a specific role when formulating an ellipse in polar coordinates.
Using the directrix in combination with the focus, it structures the equation of the ellipse. Here's how it plays into the equation:

• The distance from any point on the ellipse to the focus and to the directrix helps define the path of the ellipse.
• The conic's polar equation incorporates this directrix to express the ellipse mathematically.

In polar equations, the directrix is pivotal in establishing the parameters of the ellipse, allowing us to constructively visualize its orientation and shape.