Problem 23 - Algebra and Trigonometry

Question

(a) Find the eccentricity and directrix of the conic \(r=1 /(4-3 \cos \theta)\) and graph the conic and its directrix. (b) If this conic is rotated about the origin through an angle \(\pi / 3,\) write the resulting equation and draw its graph.

Step-by-Step Solution

Answer

(a) Eccentricity is 3, and directrix is \(x = -\frac{1}{3}\). Graph a hyperbola with this directrix. (b) Rotate using the transformation equations for \(\pi/3\) and graph the result.

Step 1: Identify the Nature of the Conic

The given equation is in the form of polar coordinates: \( r = \frac{ed}{1 - e \cos \theta} \). Here, \( ed = 1 \) and the given equation is \( r = \frac{1}{4 - 3 \cos \theta} \). By comparing, we can say \( e = 3 \) and \( ed = 1 \), hence \( d = \frac{1}{3} \). Since \( e > 1 \), this conic is a hyperbola.

Step 2: Calculate the Eccentricity

In the equation \( r = \frac{1}{4 - 3 \cos \theta} \), the coefficient of \( \cos \theta \) is 3. This represents the eccentricity of the conic. Therefore, the eccentricity \( e \) is 3.

Step 3: Find the Directrix

Using the value of \( ed = 1 \), and \( e = 3 \), we can find the directrix as \( d = \frac{1}{3} \). The directrix for a hyperbola in polar form with \( \cos \theta \) is given by \( x = -\frac{1}{3} \).

Step 4: Sketch the Graph of the Conic

Graph a hyperbola centered at the origin with the directrix \( x = -\frac{1}{3} \). Ensure that the axis along which \( \cos \theta \) varies is the horizontal axis (x-axis).

Step 5: Rotate the Conic

To rotate the conic by an angle \( \pi/3 \) about the origin, use the transformation: \( x' = x \cos(\pi/3) - y \sin(\pi/3) \) and \( y' = x \sin(\pi/3) + y \cos(\pi/3) \). For the transformed equation, substitute these expressions in place of \( x \) and \( y \).

Step 6: Write the Transformed Equation

After performing the substitution, simplify the terms to provide the new equation of the conic. The polar form will change, and several terms will come about from the trigonometric identities of \( \sin(\pi/3) \) and \( \cos(\pi/3) \).

Step 7: Graph the Rotated Conic

Graph the new equation resulting from the rotation to visualize how the hyperbola has been rotated in the plane.

Key Concepts

Eccentricity

When we talk about conic sections like ellipses, parabolas, or hyperbolas, eccentricity is a very important concept. It is a measure that helps us understand how much a conic differs from being a circle. It is represented by the letter "e".

• If 0 < e < 1, the conic is an ellipse.
• If e = 1, the conic is a parabola.
• If e > 1, as in our exercise, the conic is a hyperbola.

In our exercise, the equation given is in polar form as \[ r = \frac{1}{4 - 3 \cos \theta}\]When we compare this with the standard form of a conic in polar coordinates, we find that the eccentricity \( e \) is 3. This confirms the nature of our conic as a hyperbola. Eccentricity simply tells us that our hyperbola is more elongated than a ellipse would be. Understanding eccentricity helps in sketching and transforming the conics in different planes.

Hyperbola

A hyperbola is a type of conic section that looks like two mirror-curved lines. For hyperbolas, the eccentricity \( e \) is greater than 1. This means they have a very distinct shape compared to circles or ellipses.

In the context of the conic equation \[ r = \frac{1}{4 - 3 \cos \theta}\]we determine it to be a hyperbola because the eccentricity is 3, which is greater than 1. Hyperbolas have two branches that mirror each other, and each branch approaches two asymptotic lines but never intersects them. In the polar coordinate system, such a hyperbola is centered at the origin, making graphing straightforward once the properties are known.

To graph a hyperbola, you determine its vertices, axes, and asymptotes. The center for this hyperbola is at the origin, and its directrix is the line \( x = -\frac{1}{3} \). The directrix is pivotal for understanding the distance relations that define the hyperbola. Hyperbolas can be used in fields ranging from astronomy to physics to describe orbits and waves.

Polar Coordinates

The polar coordinate system is different from the Cartesian coordinate system. Instead of using \( x \) and \( y \) to determine a point's position, polar coordinates use a radius \( r \) (the distance from the origin) and an angle \( \theta \) (the angle from the positive x-axis). This system is very handy for graphing conics like circles, ellipses, and hyperbolas that have a central point.

Given the equation \[ r = \frac{1}{4 - 3 \cos \theta}\]we see that it's crafted in polar form, showing how polar coordinates define conics based on their symmetry about a central point. The beauty of polar coordinates is in how naturally they capture rotations and symmetries, something very helpful in problems involving conics.Transforming and rotating conics in the polar coordinate system involves mathematical formulas where \( r \) and \( \theta \) change according to trigonometric identities. For example, when the hyperbola is rotated, the transformation uses the relations \( x' = x \cos(\pi/3) - y \sin(\pi/3) \) and \( y' = x \sin(\pi/3) + y \cos(\pi/3) \). This makes the polar coordinate system perfect for dealing with angular rotations, making complex rotations much easier to handle.