Problem 23 - Precalculus

Question

Identify the conic and sketch its graph. $$r=\frac{3}{2+4 \sin \theta}$$

Step-by-Step Solution

Answer

The conic is an ellipse and the graphical representation would have its major axis of 4 units long, minor axis of 8/5 units long, and foci at \(r=\sqrt{16 - \left(\frac{8}{5}\right)^2}\).

Step 1: Identify the equation

First, identify the given equation. The provided equation is \(r=\frac{3}{2+4 \sin \theta}\). Notice that it is in the form of \(r = \frac{e}{1 + e \sin \theta}\) where \(e = \frac{3}{2}\) and \(p = 2\).

Step 2: Calculate the eccentricity

Calculate the eccentricity by dividing \(e\) by \(p\), i.e. \(e' = \frac{e}{p} = \frac{\frac{3}{2}}{2} = \frac{3}{4}\). Because \(e' < 1\), this is an ellipse.

Step 3: Graph the conic

To graph this ellipse, sketch it on the polar coordinate graph. The length of the major axis of the ellipse is \(a = \frac{e}{1-e'} = \frac{3}{2-3/4} = 4\), while that of the minor axis is \(b = \frac{e}{1+e'} = \frac{3}{2+3/4} = \frac{8}{5}\). The center at the origin, the vertex at r=4 (0 degree) and co-vertex at r=8/5 (90 degree). The foci located at \(c=\sqrt{a^2-b^2}=r=\sqrt{16 - \left(\frac{8}{5}\right)^2}\). Plot these key components to sketch the ellipse.

Key Concepts

Eccentricity of Conics

Eccentricity is a crucial concept when exploring conics, such as ellipses, hyperbolas, and parabolas, in both Cartesian and polar coordinates. It is a measure of how much a conic section deviates from being circular – with a circle having an eccentricity of 0 and a parabola having an eccentricity of 1. For an ellipse, which is the conic section in our textbook exercise, the eccentricity value lies between 0 and 1.

Eccentricity is denoted as \( e \) and is calculated differently depending on the type of conic. For an ellipse in polar form \( r = \frac{e}{1 + e' \sin \theta} \), the eccentricity \( e' \) is obtained by dividing the numerator by the focal parameter \( p \). In our case, we got \( e' = \frac{3}{4} \), indicating that our graph will be an ellipse since \( e' < 1 \). Understanding eccentricity is not just about calculating it but also about interpreting its value to predict the shape of the conic section.

In the case of the textbook exercise, the lower eccentricity suggests that our ellipse will be less stretched out, being closer to a circular shape, compared to ellipses with eccentricities approaching 1. When graphing, eccentricity helps us determine the distances of the foci from the center and the shape of the ellipse.

Polar Coordinate System

The polar coordinate system is an alternative to the Cartesian coordinate system commonly used in mathematics and engineering. It represents a point in the plane by a distance from a reference point, called the origin, and an angle relative to a reference direction.

In the polar coordinate system, the position of a point is given by a pair \((r, \theta)\), where \(r\) is the radius or the distance from the origin to the point, and \(\theta\) is the angle in radians or degrees, measured from the positive x-axis. It's particularly useful in situations where relationships are more naturally expressed in terms of distances and angles, or where symmetry about a point is present, as with conic sections.

When working with equations like that in our textbook exercise, where a conic is described using polar coordinates, we must consider both the radius and angle to properly graph the figure. Each value of \(r\) is determined by substituting the corresponding angle \(\theta\) into the equation, allowing us to plot points and sketch the curve. The polar system is especially advantageous for graphing conics because the equations often become simpler and more intuitive.

Sketching Ellipses

Sketching ellipses in polar coordinates involves understanding the parameters of the ellipse and how they relate to its shape and orientation. An ellipse's major and minor axes, foci, and center are all critical components.

Following the steps provided in the textbook solution, we identify that the given equation is in the form that indicates an ellipse once the eccentricity value has been found to be less than 1. Using this information, we can then determine the lengths of the major and minor axes, guided by the equation's parameters. In practice, these measurements define the 'width' and 'height' of the ellipse, with the major axis being the longest diameter through the center.

With the axes lengths, foci, and center at hand, we can then plot these key reference points onto the polar graph. Starting from the origin, we move out to the vertex at the end of the major axis, in our exercise at \( r = 4 \) when \( \theta = 0 \) degrees, and plot the co-vertex at the end of the minor axis, at \( r = \frac{8}{5} \) when \( \theta = 90 \) degrees. By smoothly connecting these points and ensuring the curve mirrors on both sides of each axis, we faithfully sketch out the ellipse. The foci, calculated as \( c = \sqrt{a^2 - b^2} \), are particularly useful if the ellipse were to be drawn with string and pin method, which can also aid in achieving an accurate curve.

When sketching ellipses, the symmetry and proportionality are key—it's important to maintain a consistent shape relative to the axes. Using polar coordinates can significantly simplify the process of sketching, especially ellipses, whose equations can involve trigonometric functions that are easily managed in this system.