Problem 23 - Algebra and Trigonometry

Question

A polar equation of a conic is given. (a) Show that the conic is an ellipse, and sketch its graph. (b) Find the vertices and directrix, and indicate them on the graph. (c) Find the center of the ellipse and the lengths of the major and minor axes. $$ r=\frac{12}{4+3 \sin \theta} $$

Step-by-Step Solution

Answer

The conic is an ellipse, with vertices at r=12/7, 12; center is at the origin (polar), and major axis = 6, minor axis = \( 2b = \frac{3\sqrt{7}}{2} \).

Step 1: Identify Conic Type

The given polar equation is \( r = \frac{12}{4 + 3 \sin \theta} \). To identify the conic section, we use its standard form \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 + e \cos \theta} \), where \( e \) is the eccentricity. Rewriting the given equation, we have \( r = \frac{12/4}{1 + (3/4)\sin \theta} \). Therefore, the eccentricity \( e = \frac{3}{4} \). Since \( e < 1 \), the conic is an ellipse.

Step 2: Sketch the Graph

Knowing that the conic is an ellipse, we sketch its graph using the equation \( r = \frac{12}{4 + 3 \sin \theta} \). The ellipse is oriented such that its longer axis is perpendicular to the line across the polar plane from \( \theta = \frac{5\pi}{2} \) to \( \theta = \frac{\pi}{2} \).

Step 3: Find Vertices and Directrix

The vertices occur when the function \( r \) reaches its maximum and minimum. Solve \( \sin \theta = 1 \) and \( \sin \theta = -1 \). Hence, when \( \sin \theta = 1, r = \frac{12}{4 + 3} = \frac{12}{7} \), and when \( \sin \theta = -1, r = \frac{12}{4 - 3} = 12 \). These correspond to the vertices. The directrix \( d \) can be found with \( e = c/a \) and is perpendicular to the major axis. Since \( ac = 12 \), rearranging gives \( d = \frac{ed}{1 + e \sin \theta}\) when \( r = \frac{12}{4 + 3 \sin \theta} \).

Step 4: Find Center and Axis Lengths

For an ellipse in polar form \( r = \frac{ed}{1+e\sin \theta} \), when eccentricity \( e = \frac{3}{4} \) and \( a = \frac{12}{4} = 3 \), so the center is at the polar origin. The lengths for the major axis are \( 2a = 6 \), while for the semi-major axis, it's \( 3 \). From \( b^2 = a^2(1-e^2) \), \( b = \sqrt{9\times(1-(3/4)^2)} = \sqrt{9\times(7/16)} = \frac{3\sqrt{7}}{4} \), so the minor axis is \( b = \frac{3\sqrt{7}}{4} \) and \( 2b = \frac{3\sqrt{7}}{2} \).

Key Concepts

Conic Sections

Conic sections are shapes created by the intersection of a plane and a double-napped cone. These sections come in several types, including circles, ellipses, parabolas, and hyperbolas. Each type of conic section has distinct properties and equations that define its shape.

• **Circles**: Perfectly round shapes represented by equations where the eccentricity ( e ) is 0.
• **Ellipses**: Oval shapes with an eccentricity between 0 and 1.
• **Parabolas**: Open curves with an eccentricity exactly equal to 1.
• **Hyperbolas**: Parabolas' cousins, with an eccentricity greater than 1.

Understanding these sections helps in the study of geometry, providing insights into different physical phenomena and architectural designs. Conics can be represented not only in Cartesian coordinates but also in polar coordinates, offering different perspectives and solutions to geometric problems.

Ellipse

An ellipse is a special type of conic section that's like a stretched circle. It is defined as the set of all points where the sum of the distances to two fixed points (called foci) is constant. This "oval" shape can be thought of as squeezing a circle until it has a longer axis, called the major axis, and a shorter one, called the minor axis. Ellipses have several key properties:

• **Vertices** - Points where the ellipse is widest. These points are on the major axis.
• **Center** - The point right in the middle of the ellipse.
• **Focus (Plural: Foci)** - Two special points inside the ellipse.
  

In our exercise, it's important to recognize an ellipse from its polar equation and understand the locations of the vertices and axes through given formulas. Identifying these helps in depicting the ellipse's sketch and describing its geometric orientation.

Eccentricity

Eccentricity ( e ) is a measure of how much a conic section deviates from being circular. It helps us distinguish between different types of conic sections which are defined by their unique eccentricity values.

• **Eccentricity of zero**: The conic is a circle. Perfectly round.
• **0 < Eccentricity < 1**: The conic is an ellipse. More elliptical as eccentricity approaches 1.
• **Eccentricity of one**: The conic is a parabola. Opens infinitely in one direction.
• **Eccentricity greater than one**: The conic is a hyperbola. The curve opens in two opposite directions.

For our problem, the eccentricity is \( e = \frac{3}{4} \), which tells us the conic is an ellipse. Understanding eccentricity is crucial since it dictates the shape's proportions and mathematical behavior.

Polar Equations

Polar equations describe conic sections using polar coordinates instead of the usual Cartesian coordinates. In polar coordinates, points are defined based on their distance from a central point (the origin) and the angle from a reference line.The general form of a polar equation for conics is:\[ r = \frac{ed}{1 + e \sin \theta} \text{ or } r = \frac{ed}{1 + e \cos \theta} \]In this form:

• \( r \) represents the distance from the origin to a point on the conic.
• \( \theta \) is the angle from the polar axis.
• \( e \) is the eccentricity defining the conic type.
• \( d \) is a constant related to the distance of the directrix from the pole.

For an ellipse, these equations adapt to the shape's orientation and eccentricity, allowing for plotting the curve accurately. In our specific problem, the polar equation \( r = \frac{12}{4 + 3 \sin \theta} \) neatly fitting this form helped identify the shape as an ellipse and facilitated sketching its graph.