Problem 3 - Algebra and Trigonometry

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 \( r = \frac{2}{1 - \frac{2}{3}\cos\theta} \).

Step 1: Understand the relationship

For a conic with its focus at the origin in polar coordinates, the general equation is given by \( r = \frac{ed}{1 \pm e\cos\theta} \). Since we are given an ellipse, the eccentricity \( e \) is less than 1. For our problem, \( e = \frac{2}{3} \).

Step 2: Identify the conic type and parameters

We know we have an ellipse since the given eccentricity \( e = \frac{2}{3} \) is less than 1. The directrix of the conic is a vertical line, \( x = 3 \). In polar coordinates, this influences the form of the equation.

Step 3: Substitute known values into the polar equation

Plug \( e = \frac{2}{3} \) and directrix \( d = 3 \) into the general polar equation. The equation becomes \( r = \frac{\left(\frac{2}{3}\right) \cdot 3}{1 \pm \frac{2}{3}\cos\theta} \).

Step 4: Simplify the equation

Simplifying \( r = \frac{2 \times 3}{3 \pm 2 \cos\theta} \), we find \( r = \frac{2}{1 \pm \frac{2}{3}\cos\theta} \). Since the directrix is \( x = 3 \), we use \( 1 - e\cos\theta \) (because the directrix is positive) to finalize the equation: \( r = \frac{2}{1 - \frac{2}{3}\cos\theta} \).

Step 5: Write the final polar equation

The polar equation of the ellipse is \( r = \frac{2}{1 - \frac{2}{3}\cos\theta} \).

Key Concepts

Eccentricity and Conic Sections

Eccentricity is a key concept when dealing with conic sections such as ellipses, parabolas, and hyperbolas. It essentially measures how much a conic section deviates from being circular. A circle, having no deviation, has an eccentricity of 0.

For conic sections, such as the ellipse found in this exercise, the eccentricity is a value between 0 and 1. This means the farther it is from zero, the more elongated the ellipse is. In the provided exercise, the eccentricity is given as \(\frac{2}{3}\). This indicates a moderately stretched ellipse, as it is not close to 0 but also not approaching 1.

Understanding eccentricity helps in identifying and distinguishing conic sections:

• Ellipse: \(0 < e < 1\)
• Parabola: \(e = 1\)
• Hyperbola: \(e > 1\)

Recognizing these properties allows for effective determination and construction of the conic's equation in polar and other forms.

Ellipse Properties

Ellipses are fascinating geometric shapes characterized by specific properties. One of the key features that distinguish an ellipse is its two focal points or foci. In a standard setting, these foci are symmetrically placed along the major axis, which is the longest diameter of the ellipse.

All points on an ellipse have a constant sum of distances to these two focal points. This unique property is fundamental to the shape and definition of an ellipse. It is precisely what causes the elliptical lobe to have its characteristic "oval" shape.

In polar coordinates, where one focus is at the origin, the ellipse's properties slightly transform. The eccentricity defines how stretched the ellipse is along the major axis, and this centers on the polar axis. Parameters like the directrix also come into play, affecting the ellipse's specific alignment and direction.

Mathematically, the formula \(r = \frac{ed}{1 - e\cos\theta}\) serves to describe the shape and orientation of an ellipse in polar form, with the values of \(e\) and \(d\) specifically designing the ellipse's layout and skew.

Focus and Directrix Relationship

The relationship between a conic section's focus and its directrix is critical in defining its mathematical equations and fundamental geometry. For a conic section like the ellipse in the exercise, this relationship is expressed through polar coordinates, where the focus is at the origin.

The significance of the focus and directrix structure is in their ability to establish a proportional relationship that defines the conic. The eccentricity acts as a scaling factor, determining how far the directrix is from the focus regarding the curvature and shape of the conic section.

In polar coordinates, the directrix is a critical line in calculations that aids in forming the polar equation. This positional alignment directly affects the equation's form and the labeled constants. The polar equation \(r = \frac{ed}{1 - e\cos\theta}\) is developed by considering the directrix's position alongside the eccentricity, which adjusts the equation to accommodate the directrix \(x = 3\) as noted in this particular problem.

Thus, understanding the focus-directrix relationship not only unveils the beauty behind these geometric figures but also helps in interpreting and solving problems involving conic sections.