Problem 52 - Thomas Calculus

Question

Graph the lines and conic sections in Exercises \(47-56\) $$ r=8 /(4+\sin \theta) $$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Understanding the Polar Equation

The given equation is in polar form: \[r = \frac{8}{4 + \sin \theta}\].This represents a type of conic section. In polar coordinates, conic sections can be represented using equations of the form \( r = \frac{ed}{1 + e\sin(\theta)} \) or \( r = \frac{ed}{1 + e\cos(\theta)} \), where \(e\) is the eccentricity, and \(d\) is the directrix.

Step 2: Identifying the Conic Section

Compare \(r = \frac{8}{4 + \sin \theta}\) with the standard polar conic section form: \[ r = \frac{ed}{1 + e\sin\theta} \].Here, the equation can be rewritten as \( r = \frac{2}{1 + \frac{1}{4}\sin\theta} \), suggesting that \( ed = 2 \), \( e = \frac{1}{4} \), and \( d = 8 \). Since \( e \) is less than 1, the conic section is an ellipse.

Step 3: Graphically Plotting the Ellipse

To graph the ellipse, consider various values of \(\theta\): 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\). Substitute these values into the equation to find corresponding \(r\) values:- When \(\theta = 0\), \(r = \frac{8}{4+0} = 2\).- When \(\theta = \frac{\pi}{2}\), \(r = \frac{8}{4+1} = \frac{8}{5}\).- When \(\theta = \pi\), \(r = \frac{8}{4+0} = 2\).- When \(\theta = \frac{3\pi}{2}\), \(r = \frac{8}{4-1} = \frac{8}{3}\).Plot these points on polar graph paper and connect them smoothly to form an ellipse.

Key Concepts

Conic Sections

Conic sections are a crucial concept in mathematics that provides insight into different types of curves that you can generate by intersecting a plane with a cone. There are four basic types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each type has its unique properties and equations.

• Circle: An ellipse with an eccentricity of 0, meaning it is perfectly round.
• Ellipse: An oval-shaped curve with eccentricity values between 0 and 1.
• Parabola: A curve where the eccentricity is exactly 1. It looks like a stretched U-shape.
• Hyperbola: A curve with an eccentricity greater than 1, resembling two disconnected curves facing away from each other.

Understanding conic sections is essential as they appear in various fields like physics, engineering, and architecture. Whether you are calculating the path of a satellite or designing a bridge, conic sections offer valuable models and solutions.

Eccentricity

Eccentricity is a measure that helps identify the shape of a conic section. It gives you an idea of how much a conic section deviates from being circular.
- For circles, the eccentricity (\( e \)) is 0, which means no deviation.- For ellipses, eccentricity ranges between 0 and 1.- A value of 1 characterizes a parabola.- Values greater than 1 describe hyperbolas.
In our example equation \( r = \frac{8}{4 + \sin \theta} \), the eccentricity \( e \) is \( \frac{1}{4} \). Since \( e \) is less than 1, the shape formed is an ellipse. Eccentricity not only tells us what type of conic section we are dealing with but also provides information on the shape and dimensions of the figure. This small number signifies that the ellipse is fairly close to being a circle.

Polar Equation

Polar equations provide a systematic way to represent curves in the polar coordinate system. This system is different from the traditional Cartesian coordinate system and is particularly useful for certain types of geometrical shapes, such as circles, spirals, and, importantly, conic sections.
A polar equation has the form \( r = \frac{ed}{1 + e\sin\theta} \) or \( r = \frac{ed}{1 + e\cos\theta} \), where:

• \( r \) is the radius or distance from the origin to a point on the curve.
• \( \theta \) is the angle from the polar axis.
• \( e \) is the eccentricity, telling us the type of conic section.
• \( d \) is the directrix, a line used to construct and define the curve.

In the exercise given, the polar equation \( r = \frac{8}{4 + \sin \theta} \) simplifies to \( r = \frac{2}{1 + \frac{1}{4}\sin\theta} \), showing it is an ellipse due to its specific form and values.

Graphing Ellipse

Graphing an ellipse in polar coordinates involves converting polar equations into a visual format that represents the shape of the curve on a plane. To effectively plot an ellipse, follow these straightforward steps:
1. Identify key values: Find various \( \theta \) values that are easy to work with (like \( 0, \frac{\pi}{2}, \pi, \) and \( \frac{3\pi}{2} \)). 2. Calculate \( r \) for different \( \theta \): Substitute each \( \theta \) into the polar equation to determine the corresponding \( r \).3. Plot points: Use the \( r \) and \( \theta \) pairs to plot points on polar graph paper.4. Connect points: Gently stitch the points together to show the shape of the ellipse.
In our exercise, specific \( r \) values like 2, \( \frac{8}{5} \), 2, and \( \frac{8}{3} \) are obtained based on our chosen \( \theta \) values. Connect these smoothly, and an elegant ellipse appears on your graph. Utilizing polar coordinates for graphing is intuitive once you get the hang of it, providing beautiful and precise curves.