Question

For the following exercises, sketch the graph of each conic. $$ r=\frac{3}{-4+2 \sin \theta} $$

Short answer

The graph is an ellipse with \(e = \frac{1}{2}\), centered at the pole.

Step-by-step solution

Identify the Conic Section Form

The given polar equation is \(r = \frac{3}{-4 + 2 \sin \theta}\). We recognize this as a conic section in polar coordinates, which has the general form \(r = \frac{ed}{1 + e\sin\theta}\). Upon comparison, we identify the eccentricity \(e\) and constant \(d\).

Rewrite in Standard Conic Form

Rewrite the given equation in the form \(r = \frac{ed}{1 + e\sin\theta}\) to better analyze it. Thus we have \(r = \frac{3}{-4 + 2 \sin \theta} = \frac{3/-4}{1 - (-\frac{2}{-4})\sin\theta}\). Simplifying, \(d = \frac{3}{-4}\) and \(e = -\frac{2}{-4} = \frac{1}{2}.\)

Determine the Type of Conic

Since we found \(e = \frac{1}{2}\), which is less than 1, the conic is an ellipse. The negative denominator initially and switching signs in standard form indicates that this is a conic opening in the opposite direction.

Find the Directrices of the Ellipse

For an ellipse, the directrix \(d\) needs to be calculated, which in this case is already given as \(- \frac{3}{4}\). This confirms that the ellipse has a horizontal orientation.

Graph the Conic

Using the information that we have an ellipse centered at the pole, we can sketch the graph. Since \(e < 1\), it is less eccentric and more circular-like. Plot key points considering symmetry and the principal axis with negative focal direction.

Key concepts

Polar Coordinates

Polar coordinates offer a unique way to describe a point in a plane using a distance and an angle. Unlike the Cartesian system, which uses the x and y axes, polar coordinates use two components:

• The radius ( "), often denoted by \(r\), measures the distance from a central point known as the pole (analogous to the origin in the Cartesian system).
• The angle (\(\theta\)), measured from a fixed direction (usually the positive x-axis), determines the direction of the point from the pole.

This system is particularly useful in scenarios where symmetry about a point simplifies the mathematics, such as in circular and spiral structures.
In the given context of the exercise, the equation \(r = \frac{3}{-4 + 2 \sin \theta}\) is expressed in polar coordinates, indicating how the size of the radius changes as the angle \(\theta\) changes. This form makes it easier to analyze the nature of the conic section, which in this case, involves an ellipse.

Eccentricity

Eccentricity is a key concept in understanding the shape and nature of conic sections. It is a non-negative real number that indicates how much a conic deviates from being circular. The value of eccentricity \(e\) determines the type of conic section:

• If \(e = 0\), the conic is a circle.
• If \(0 < e < 1\), the conic is an ellipse.
• If \(e = 1\), the conic is a parabola.
• If \(e > 1\), the conic is a hyperbola.

In our exercise, we calculated an eccentricity of \(e = \frac{1}{2}\). Since this value is between 0 and 1, it confirms that the conic section is an ellipse. The smaller the eccentricity, the more the ellipse resembles a circle. This concept helps in determining the geometrical properties of the figure and understanding the degree of its elongation.

Ellipse

An ellipse is a smooth, closed curve that results from the conic section formed when a plane cuts through a cone at an angle. Its defining characteristic is that the total distance from any point on the ellipse to two fixed points, called foci, is constant.
The standard polar equation for an ellipse is \(r = \frac{ed}{1 + e\sin\theta}\), with the eccentricity \(e < 1\).
Ellipses have several distinct features:

• The major and minor axes determine the shape's dimensions. The major axis is the longest diameter of the ellipse, while the minor axis is the shortest.
• Ellipses have symmetry across these axes, making them aesthetically appealing and useful in real-world applications like planetary orbits.

In the given exercise, the form \(r = \frac{3}{-4 + 2 \sin \theta}\) was converted to determine the ellipse's parameters, revealing its horizontal orientation due to the negative denominator. This transformation lets us visualize and graph the ellipse effectively.

Graphing Conic Sections

Graphing conic sections involves plotting the set of all points satisfying a given equation. Each conic type — circles, ellipses, parabolas, and hyperbolas — has its own characteristics to consider during the graphing process.
For ellipses, as derived from the eccentricity and polar form, key points need to be plotted considering both the principal axis and symmetry. In our exercise, after identifying the conic as an ellipse, the focus shifts to plotting it correctly:

• Determine the general shape by considering the eccentricity and orientation, indicated as horizontal here due to trigonometric symmetry.
• Plot key intercepts and smooth the curve, ensuring it resembles a flattened circle, indicative of the lower eccentricity value.
• As it’s centered at the pole, ensure the graph encapsulates all relevant focal geometry features.

This methodical approach simplifies understanding and visualizing the complex ideas behind polar equations and their manifestations as conic sections in a graph.