Problem 26 - Precalculus: Mathematics for Calculus

Question

A polar equation of a conic is given. (a) Show that the conic is a hyperbola, and sketch its graph. (b) Find the vertices and directrix, and indicate them on the graph. (c) Find the center of the hyperbola, and sketch the $$r=\frac{10}{1-4 \sin \theta}$$

Step-by-Step Solution

Answer

(a) The conic is a hyperbola; (b) Vertices are at \((0, -10)\) and \((0, 10)\); (c) Center is at \((0, 0)\).

Step 1: Identify the Type of Conic

The given equation is \( r = \frac{10}{1-4 \sin \theta} \). A polar equation of a conic takes the form \( r = \frac{ed}{1-e \sin \theta} \) or \( r = \frac{ed}{1-e \cos \theta} \). Hence, by comparing, we notice that the eccentricity \( e = 4 \). Since \( e > 1 \), the conic is a hyperbola.

Step 2: Sketch the Graph

To sketch a hyperbola in polar coordinates, note that the equation is of the form \( r = \frac{ed}{1-e \sin \theta} \), suggesting the directrix is vertical. The exponents and negative sine indicate that one focus is at the pole, and the conic extends in the direction opposite to the pull of the sine terms. Plot the symmetric branches of the hyperbola by calculating values for various angles.

Step 3: Find the Vertices

The vertices of the hyperbola occur when \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \). Substitute these values into the equation and solve for \( r \). When \( \theta = \frac{\pi}{2} \), \( r = -10 \), and when \( \theta = \frac{3\pi}{2} \), \( r = 10 \). Thus, the vertices are at \( (0, -10) \) and \( (0, 10) \).

Step 4: Determine the Directrix

For \( r = \frac{ed}{1-e \sin \theta} \), the directrix is a line perpendicular to the polar axis. Given \( e = 4 \), the directrix is found by using the formula \( r = \pm \frac{d}{1 - e} \) when \( \theta = 0 \) or \( \theta = \pi \). Hence, one possible directrix is at \( \frac{1-4}{-4}\), so \( y = -\frac{1}{4} \) passing through the pole.

Step 5: Find the Center

Since the focus of this polar equation is at the pole, the center is also at the pole, \( (0, 0) \). Marking the vertices and directrix on the graph will aid in accurately representing the hyperbola.

Key Concepts

Hyperbola

A hyperbola is a type of conic section that is defined by its distinctive geometric shape, forming an open curve with two distinct branches. Each branch appears mirror imaged if placed against a vertical axis. In the context of polar coordinates, a hyperbola can still be identified by evaluating the equation given. If the eccentricity, denoted as \( e \), is greater than 1, then the conic section is confirmed to be a hyperbola. For example, in a given polar equation like \( r = \frac{10}{1-4 \sin \theta} \), the eccentricity \( e = 4 \) indicates a hyperbola because it exceeds one. Understanding this property is crucial for sketching and analyzing the behavior of the graph.

Conic Sections

Conic sections describe the curves obtained by slicing a cone with a plane at different angles. They include circles, ellipses, parabolas, and hyperbolas. These curves have unique properties that make them interesting and valuable in both mathematics and physics. In polar coordinates, these sections can be expressed in a specific polar equation form: \( r = \frac{ed}{1-e \sin \theta} \) or \( r = \frac{ed}{1-e \cos \theta} \). The parameter \( e \) denotes the eccentricity, which determines the specific type of conic section. By analyzing an equation in this form, you can determine whether the conic section is a circle, ellipse, parabola, or hyperbola based on the eccentricity value.

Eccentricity

Eccentricity is a key parameter that defines the shape of a conic section. It is represented by the letter \( e \). Depending on its value, it can classify the conic into different types:

• 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 \( e > 1 \), the conic is a hyperbola.

In polar coordinates, it is especially useful when evaluating equations of the form \( r = \frac{ed}{1-e \sin \theta} \). For the given equation, \( e = 4 \), which confirms that the conic section is a hyperbola. Insight into this property helps in visualizing the curves' structure and graphing them correctly.

Vertices of Conic

Vertices are pivotal points on a conic section that describe its main dimensions. For hyperbolas in polar equations, vertices can often be derived by evaluating the equation at particular angles, usually \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \). Substituting these angle values into the polar equation \( r = \frac{10}{1-4 \sin \theta} \), we determine that:

• At \( \theta = \frac{\pi}{2} \), the radius \( r = -10 \).
• At \( \theta = \frac{3\pi}{2} \), \( r = 10 \).

Thus, the vertices are located at \( (0, -10) \) and \( (0, 10) \). Knowing the positions of the vertices is essential for drawing the hyperbola and understanding its spatial orientation.

Directrix

The directrix of a conic section is a line used in conjunction with the eccentricity to help define and draw the shape. In the polar equation of a conic section, the directrix can be found using the notion of eccentricity. For example, in the formula \( r = \frac{ed}{1-e \sin \theta} \), if the eccentricity \( e = 4 \), it is pertinent as it influences the position of the directrix relative to the focus at the pole. For some conic forms, the directrix can parallel or align perpendicularly to the polar axis depending on the situational needs of the conic type. Using the equation in the steps, with \( r = \frac{10}{1-4 \sin \theta} \), a vertical directrix can be derived, pointing to lines such as \( y = -\frac{1}{4} \) which run past the defined focus. Establishing the directrix ensures you sketch the hyperbola or other sections correctly in graphs.