Problem 40 - Precalculus Mathematics for Calculus

Question

A polar equation of a conic is given. (a) Find the eccentricity and the directrix of the conic. (b) If this conic is rotated about the origin through the given angle \(\theta\), write the resulting equation. (c) Draw graphs of the original conic and the rotated conic on the same screen. $$r=\frac{9}{2+2 \cos \theta} ; \quad \theta=-\frac{5 \pi}{6}$$

Step-by-Step Solution

Answer

The conic is a parabola with eccentricity 1 and directrix \( r = 4.5 \). After rotation, the equation is \( r = \frac{9}{2 - \sqrt{3}\cos\theta - \sin\theta} \).

Step 1: Identify the Type of Conic

The general form of a conic in polar coordinates is \( r = \frac{ed}{1 + e\cos\theta} \) or \( r = \frac{ed}{1 + e\sin\theta} \). Comparing with the given equation \( r = \frac{9}{2 + 2\cos\theta} \), we identify this as \( r = \frac{9}{2(1 + \cos\theta)} \). This fits the format of the conic equation with \( e = 1 \), indicating the conic is a parabola.

Step 2: Compute the Eccentricity

From the identified conic form, \( e = 1 \). The eccentricity \( e = 1 \) implies that the conic is a parabola.

Step 3: Find the Directrix

Since the equation has the form \( r = \frac{ed}{1 + e\cos\theta} \) with \( e = 1 \), the directrix formula is \( r = \frac{d}{2} \), leading to \( d = 9 \). Therefore, the directrix is the line \( r = 4.5 \).

Step 4: Identify the Rotation Effect

When rotating a conic equation \( r = \frac{ed}{1 + e\cos\theta} \) through an angle \( \theta = -\frac{5\pi}{6} \), replace \( \theta \) with \( \theta + \frac{5\pi}{6} \). The new form becomes \( r = \frac{9}{2 + 2\cos(\theta + \frac{5\pi}{6})} \).

Step 5: Simplify the Rotated Equation

Use the angle addition formula, \( \cos(\theta + \frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}\cos\theta - \frac{1}{2}\sin\theta \), to rewrite the rotated function. Substitute this into the equation to get \( r = \frac{9}{2 - \sqrt{3}\cos\theta - \sin\theta} \).

Step 6: Graph the Conics

To graph the original and rotated conics, plot \( r = \frac{9}{2 + 2\cos\theta} \) for the original conic and \( r = \frac{9}{2 - \sqrt{3}\cos\theta - \sin\theta} \) for the rotated conic. These graphs should reflect the original parabola's position and its rotation through the specified angle.

Key Concepts

Eccentricity

Eccentricity is a vital concept when studying conic sections in polar and Cartesian coordinates. It's a measure of how much the shape of the conic section deviates from being circular.

• If the eccentricity, denoted as \( e \), equals \( 0 \), the conic is a circle.
• If \( 0 < e < 1 \), it indicates an ellipse.
• When \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), it's a hyperbola.

For the given polar equation \( r = \frac{9}{2 + 2 \cos \theta} \), it conforms to the form \( r = \frac{ed}{1 + e \cos \theta} \) where the eccentricity \( e = 1 \). Hence, the conic section is a parabola. The eccentricity of \( 1 \) ensures the parabolic nature, where the path focuses outward equally from a fixed point.

Directrix

The directrix plays a crucial role in defining conic sections. In simple terms, a directrix is a straight line used in the description of a curve or surface. It is particularly meaningful in understanding how conics like parabolas are shaped and oriented.
For conic sections defined in polar coordinates, the directrix can be calculated using the relationship to eccentricity. Using the provided form \( r = \frac{ed}{1 + e \cos \theta} \) and knowing \( e = 1 \), the equation simplifies to \( r = \frac{d}{2} \).
In our particular case, by equating \( rac{ed}{2} = 4.5 \), we find that the directrix is at \( r = 4.5 \). This value indicates where the line is positioned relative to the origin, guiding the parabolic path in a consistent and predictable manner. For parabolas, the focus and the directrix are extremely important as each point on the parabola is equidistant from both the directrix and the focus.

Conic Sections

Conic sections are shapes created as a plane intersects a double napped cone. They are defined by their eccentricity and include circles, ellipses, parabolas, and hyperbolas. In polar coordinates, these sections can be expressed in a compact form, which aids in visualization and computation.

• Circles have an eccentricity of \( e = 0 \).
• Ellipses have an eccentricity \( 0 < e < 1 \).
• Parabolas, like the given equation \( r = \frac{9}{2 + 2 \cos \theta} \), have an eccentricity of \( e = 1 \).
• Hyperbolas occur when \( e > 1 \).

In the exercise, our focus is on parabolas, specifically defined by the provided equation. By plotting conic sections in polar coordinates, students can gain a deeper understanding of how directrices, foci, and eccentricities influence these curves' shapes and orientations. Recognizing these conics through their standard forms and geometric properties is essential in many fields, such as mathematics, physics, and engineering.