Problem 4 - Algebra and Trigonometry

Question

1–8 Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\frac{1}{2},\) directrix \(y=-4\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{2}{1 - \frac{1}{2} \sin \theta} \).

Step 1: Identify the form of the polar equation

The polar equation of a conic with focus at the origin is given by \( r = \frac{ed}{1 - e \sin \theta} \) if the directrix is vertical, and \( r = \frac{ed}{1 - e \cos \theta} \) if the directrix is horizontal. In this exercise, since the directrix is \( y = -4 \), which is horizontal, we use the form \( r = \frac{ed}{1 - e \sin \theta} \).

Step 2: Identify given parameters and eccentricity

The eccentricity \( e \) is given as \( \frac{1}{2} \), and the directrix is \( y = -4 \). Since the directrix is horizontal, \( d = 4 \) and the conic opens upwards point towards the positive y-direction. Using \( r = \frac{ed}{1 - e \sin \theta} \).

Step 3: Substitute the known values into the equation

Substitute \( e = \frac{1}{2} \) and \( d = 4 \) into the equation \( r = \frac{ed}{1 - e \sin \theta} \). This gives us \( r = \frac{ \left(\frac{1}{2}\right) \cdot 4}{1 - \left(\frac{1}{2}\right) \sin \theta} \).

Step 4: Simplify the equation

Calculate the expression: \( r = \frac{2}{1 - \frac{1}{2} \sin \theta} \). This is the polar equation of an ellipse with a focus at the origin, eccentricity \( \frac{1}{2} \), and directrix \( y = -4 \).

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. They have fascinating properties and are commonly categorized into four types: circles, ellipses, parabolas, and hyperbolas. These shapes are part of the same family and are distinguished by their unique geometric and algebraic properties.

• A **circle** is a simple closed shape where all points are equidistant from the center.
• An **ellipse** is like an elongated circle with two principal axes.
• A **parabola** looks like a symmetrical curve that mirrors on its axis.
• A **hyperbola** consists of two separate branches divergent from one another.

Conic sections are not just theoretical; they have practical applications ranging from astronomy to engineering. They are defined in several ways, one being the eccentricity and directrix, which we'll explore next.

Eccentricity

Eccentricity is a measure that determines how much a conic section deviates from being circular. It's represented by the letter **e** and is one of the most valuable properties in classifying conic sections:

• If **e = 0**, the conic is a perfect circle.
• If **0 < e < 1**, the conic is an ellipse.
• If **e = 1**, we get a parabola.
• If **e > 1**, the conic is a hyperbola.

In the original exercise, the eccentricity was given as \( \frac{1}{2} \), which indicates the conic is an ellipse. Lower eccentricity values in ellipses mean they are closer to being circular, while values closer to 1 make them more elongated. This concept of eccentricity not only helps in classifying conic sections but also in understanding their geometric behaviors and relationships.

Directrix

The directrix is a fixed line used in the definition and equation of a conic section. It often assists in constructing conics when combined with the focus and eccentricity. For a given conic section,

• A **directrix** serves as a benchmark line to which the distance of every point on the conic is compared.
• In any conic, for a point on the curve, the ratio of its distance from the focus to its distance from the directrix is constant and equals the eccentricity: \( e \).

In the exercise, the directrix is a horizontal line defined at \( y = -4 \). This specific placement influences the conic's orientation and shape in the coordinate system. Understanding the relationship between the conic and its directrix is essential for solving problems involving polar equations or translating them between different forms.