Problem 27 - Precalculus with Limits

Question

In Exercises 15-28, identify the conic and sketch its graph. \(r=\dfrac{4}{2-\cos\ \theta}\)

Step-by-Step Solution

Answer

The conic given by the polar equation \(r=\frac{4}{2-\cos\ \theta}\) is a hyperbola.

Step 1: Identify the conic

The polar equation given is in the form \(r=\frac{ep}{1-e\cos(\theta-\theta_0)}\) where \(e=2, p=2, \theta_0=0\). Please refer to the conic section general equations mentioned earlier, it can be inferred that the conic is a hyperbola because \(e>1\).

Step 2: Sketch the conic graph

Since this is a hyperbola, first draw the focal points by finding the directrix, which is \(p=\frac{4}{2-2}=2\). Then, place the focal points along the x-axis or the polar axis. Finally, plot a few points using the polar equation and sketch the hyperbola.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped right circular cone. Depending on the angle of the plane relative to the cone, different shapes are formed, categorized into four main types: circles, ellipses, parabolas, and hyperbolas.

Each conic section has its unique equation defining its shape. In polar coordinates, conic sections are represented by equations involving the radius r and the angle θ. The general form of the conic section equation in polar coordinates is r = (e·p) / (1 - e·cos(θ - θ₀)), where e is the eccentricity, p is the semi-latus rectum, and θ₀ is the orientation of the conic.

The eccentricity e is a key factor in determining the conic section's type:

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

Understanding these core principles allows students to move forward in identifying conic sections from their polar equations.

Hyperbola Sketch

When it comes to graphing a hyperbola from its polar equation, such as r = 4 / (2 - cos(θ)), where the eccentricity e > 1, it's pivotal to follow a step-by-step approach:

First, identify the conic type by checking the eccentricity. Here, e = 2, indicating a hyperbola. Next, determine the directrix and focal points. The directrix helps establish the distance at which the curve approaches but never reaches, which is obtained by the semi-latus rectum p.

The focal points of a hyperbola in polar coordinates lie along the polar axis, and one of them is at the origin. In this specific example, since θ₀ = 0, the transverse axis of the hyperbola is aligned with the polar axis. To sketch the hyperbola:

• Plot the focal points based on p.
• Choose several values of θ to calculate r and plot these points.
• Draw an asymptotic box (if applicable), which is determined by the directrix and asymptotes.
• Connect these points smoothly, creating the two distinct branches of the hyperbola that mirror each other across the transverse axis.

For a more accurate sketch, additional points can be plotted and the behavior of the hyperbola near the center can be examined.

Polar Coordinates Graphing

Graphing in polar coordinates involves plotting points and curves based on their distance from the origin r and the angle θ relative to the polar axis (the positive x-axis in Cartesian coordinates). This system is particularly convenient for circular and spiral patterns and equations representing conic sections.

To graph an equation in polar coordinates:

• Select a range of θ values (typically from 0 to 2π for a full revolution).
• Compute the corresponding r values using the given polar equation.
• For each (r, θ) pair, move r units away from the origin in the direction specified by the angle θ.
• Mark the point and repeat for all selected θ values.

When graphing polar equations of conics on a grid, you may use radial lines for angles and concentric circles for radius increments to aid in plotting points accurately. The beauty of polar graphing lies in its symmetry and periodicity; recognizing these patterns can simplify drawing curves and shapes, especially conic sections.