Question

For the following exercises, sketch the graph of each conic. $$ r=\frac{15}{3-2 \cos \theta} $$

Short answer

The graph is an ellipse centered at the pole with a horizontal orientation.

Step-by-step solution

Identify the Conic Type

The given polar equation is \( r=\frac{15}{3-2 \cos \theta} \). This is in the form \( r = \frac{ed}{1 - e\cos \theta} \), which represents a conic section. Comparing this to the standard form reveals that the conic section is an ellipse, where \( e=\frac{2}{3} \), a value less than 1.

Determine the Orientation

The numerator of our equation is a constant, and the denominator involves \( \cos \theta \), which indicates that the orientation of the ellipse is horizontal. The major axis will be aligned with the horizontal line due to the presence of \( \cos \theta \) in the equation.

Find Vertex and Foci Positions

Since the conic opens horizontally along the polar axis, we must confirm the vertex positions. The vertices occur when the denominator is minimized and maximized. Substitute \( \theta = 0 \) (cosine maximum), the point is closest so \( r = \frac{15}{1} = 5 \). For \( \theta = \pi \) (cosine minimum), the point is farthest so \( r = -5 \). For \( \theta = \pi/2 \) and \( 3\pi/2 \) (cosine zero), the asymptotic value towards infinity, where radius grows without bound, indicating foci beyond limits.

Sketch the Graph

Draw a horizontal ellipse centered at the pole. The closest approach at the pole is when \( \theta = 0 \), plotting a vertex at \( (5,0) \). The extremal point at the opposite direction for \( \theta = \pi \) is contrasonic, implying symmetry across the \( \pi/2 \) and \( 3\pi/2 \). Sketch arcs of the ellipse diverging at these radians with the elongation mostly horizontal.

Key concepts

Conic Sections

Conic sections are fundamental shapes in geometry that arise from the intersection of a plane with a double-napped cone. Essentially, they include circles, ellipses, parabolas, and hyperbolas. In mathematics, these conic sections are widely studied since they have unique properties and appear frequently in various fields.

When given in polar coordinates, conic sections have different forms. The general polar equation for a conic section is \[r = \frac{ed}{1-e\cos \theta} \]where \( e \) is the eccentricity that defines the shape's nature:

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

This understanding allows us to quickly identify and categorize conic sections when dealing with equations in polar form. Recognizing these patterns helps predict the graph's shape and orientation, providing a solid foundation for graph sketching.

Ellipse

An ellipse is a particular type of conic section characterized by an eccentricity \( e \) that is greater than 0 but less than 1. In simpler terms, it looks like an elongated circle. This shape has two main axes, known as the major and minor axes.

For the equation \[r=\frac{15}{3-2 \cos \theta}\]we identified that it represents an ellipse because the eccentricity \( e = \frac{2}{3} \), which fulfills the condition \( 0 < e < 1 \).

The ellipse's major axis orientation is determined by the trigonometric function within the equation. Here, the \( \cos \theta \) term indicates a horizontal major axis. The major axis is the longest diameter, stretching across the widest part of the ellipse. Important properties of ellipses include:

• Foci: Two fixed points on the major axis, central to defining an ellipse.
• Vertices: Points on the ellipse that lie on the major and minor axes.
• Center: The midpoint termed as the average position, located at the pole in polar coordinates.

In this context, identifying vertex positions and orientation is key to sketching the ellipse accurately.

Graph Sketching

Sketching the graph of a conic section such as an ellipse involves understanding its components and using the polar equation to determine critical points. The previous polar equation\[r=\frac{15}{3-2 \cos \theta}\]gives us a clear picture of how to graph a horizontal ellipse, focusing on key attributes such as vertices and foci.

The vertices are calculated based on critical angles of \( \theta \):

• For \( \theta = 0 \), the closest point to the center is at \( r = 5 \).
• For \( \theta = \pi \), the farthest point (contrasonic) is at \( r = -5 \).
• At \( \theta = \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), the points expand infinitely, confirming foci placement beyond boundaries.

To sketch, start by drawing the center at the pole with significant points marked for vertices. Draw arcs approximating the ellipse's curvature, taking care to reflect the symmetry about horizontal major axes. It is crucial to illustrate the points and curves accurately to represent how the ellipse would appear.