Problem 44 - Thomas Calculus

Question

Sketch the parabolas and ellipses in Exercises \(37-44\) . Include the directrix that corresponds to the focus at the origin. Label the vertices with appropriate polar coordinates. Label the centers of the ellipses as well. $$r=\frac{4}{2-\sin \theta}$$

Step-by-Step Solution

Answer

The sketch is an ellipse. Center: Origin, vertices at \((2, 0)\), and \((-4, \pi)\). Directrix: \(d = 8\).

Step 1: Identify the Conic

The given equation is \( r = \frac{4}{2 - \sin \theta} \). This equation is in the form \( r = \frac{ed}{1 - e \sin \theta} \), where \( e \) is the eccentricity, and \( d \) is the directrix. Since the denominator contains \( \sin \theta \), it indicates a conic oriented vertically. Let's compare to recognize the conic type.

Step 2: Determine Eccentricity and Type

For the given conic \( r = \frac{4}{2 - \sin \theta} \), compared with \( r = \frac{ed}{1 - e \sin \theta} \), we have \( ed = 4 \) and \( 1 - e = 2 \). Solving \( 1 - e = 2 \) gives us \( e = -1 \). This indicates a parabola since the secondary requirement for forming an ellipse (\( 0 < e < 1 \)) is not satisfied.

Step 3: Adjust Equation for Standard Form

As \( e = 1 \), it appears our earlier calculation seems incorrect. Reviewing, since \( ed = 4 \) and \( 1 - e = 2 \), gives \( e = \frac{1}{2} \). Therefore, our conic becomes \( r = \frac{4}{2 - \sin \theta} \), and \( e = \frac{1}{2} \) indicates an ellipse.

Step 4: Calculate Directrix and Verify Form

Given \( e = \frac{1}{2} \) and the relationship \( ed = 4 \), we solve for \( d \): \( \left(\frac{1}{2}\right)d = 4 \Rightarrow d = 8 \). The standard form for a conic oriented this way then becomes \( r = \frac{4}{2 - \sin \theta} \). Since we have \( e = \frac{1}{2} \), this ellipse is centered at the origin.

Step 5: Sketch the Ellipse

To sketch: plot the center at the origin. The ellipse's major axis is vertical, moving with variations of \( \sin \theta \). To get a rough sketch: as \( \theta \to \frac{\pi}{2} \), \( r \to 8 \), ensuring our major axis lies along the positive y-direction.

Step 6: Label the Polar Coordinates

Identify vertices and directrix. The vertices occur when \( \theta = 0 \) and \( \theta = \pi \), translating to \( r = \frac{4}{2 - 0} = 2 \) and \( r = \frac{4}{2 - (-\sin \pi)} = 4 \). Thus, the vertices are at \((2, 0)\) and \((-4, \pi)\). Center is at the origin.

Key Concepts

Eccentricity

Eccentricity is a crucial concept in the discussion of conic sections, which helps in determining the type of conic. It is expressed by the variable \( e \). The eccentricity decides the shape of the conic section:

• For a circle, \( e = 0 \).
• In the case of an ellipse, \( 0 < e < 1 \).
• A parabola is identified by \( e = 1 \).
• And for a hyperbola, \( e > 1 \).

In our equation \( r = \frac{4}{2 - \sin \theta} \), we first solved for eccentricity by setting conditional equations derived from the general form. By solving these equations, we found \( e = \frac{1}{2} \), which indicates that the conic we are dealing with is an ellipse. Comprehending eccentricity allows us to determine how "circular" or stretched an ellipse is. The closer \( e \) is to 1, the more elongated the ellipse appears.

Directrix

The directrix of a conic section is a line used to help define and construct the conic. It correlates closely with eccentricity. Its primary role is to remain equidistant from any point on the conic to the focus. In polar coordinates, it is often defined by equations that involve eccentricity:

• For a line-bound ellipse, the directrix adapts given \( d \), related by \( ed = k \).
• Directrix acts as an anchor point, in contrast to the point-like behavior of a focus.

From our solved equation, we find the directrix by using \( ed = 4 \), with given \( e = \frac{1}{2} \), solving provides \( d = 8 \). Thus, the directrix in this context is a horizontal line at \( r = 8 \). Understanding the directrix helps us sketch accurate geometric figures, offering positional context.

Polar Coordinates

Polar coordinates are a system where the position of a point is given by its distance from a reference point (the pole, usually the origin) and the angle from a reference direction. It's expressed as \((r, \theta)\):

• \( r \) indicates the distance from the pole.
• \( \theta \) represents the angle in the polar plane.

Polar coordinates serve uniquely in dealing with curves and lines that are symmetrical about the pole. In our example, \( r = \frac{4}{2 - \sin \theta} \) conforms to this system. It reflects how \( r \) changes Kour with the angle \( \theta \), sketching the form of an ellipse centered at the origin. Transitioning between rectangular coordinates and polar coordinates is frequent in conic sections, particularly in simplifying the representation and sketching curves.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They encompass circles, ellipses, parabolas, and hyperbolas, each distinguished by its eccentricity value:

• Circle: perfect rounded figure, eccentricity zero.
• Ellipse: elongated circle, eccentricity less than one.
• Parabola: open curve, eccentricity equal to one.
• Hyperbola: two symmetrical open curves, eccentricity greater than one.

These curves have numerous applications from planetary motion to architectural designs. The equation \( r = \frac{4}{2 - \sin \theta} \) analyzed in the exercise, confirmed as an ellipse, shows how varied the conic sections can be. Mastering conic sections in polar coordinates aids in visualizing and solving geometric problems efficiently.