Problem 27 - Precalculus: Functions and Graphs

Question

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=\frac{4}{3}, \quad r \cos \theta=-3$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{12}{3 + 4\cos \theta} \).

Step 1: Understand the Problem

We need to find a polar equation that represents a conic section with given eccentricity \( e = \frac{4}{3} \) and the directrix \( r \cos \theta = -3 \). The focus of the conic is located at the pole.

Step 2: Identify the Type of Conic

The eccentricity \( e > 1 \) indicates the conic is a hyperbola. A hyperbola has eccentricity greater than 1.

Step 3: Use Conic Polar Equation Formula

For a conic with a focus at the pole, the polar equation is given by \[ r = \frac{ed}{1 + e \cos \theta} \] if the directrix is vertical at \( x = -d \). Here, \( e = \frac{4}{3} \) and the directrix equation \( r \cos \theta = -3 \) gives us \( d = 3 \).

Step 4: Substitute Values into Formula

Substitute \( e = \frac{4}{3} \) and \( d = 3 \) into the polar equation: \[ r = \frac{\left(\frac{4}{3}\right)(3)}{1 + \frac{4}{3} \cos \theta} \]

Step 5: Simplify the Expression

Simplify the equation: \[ r = \frac{4}{1 + \frac{4}{3} \cos \theta} \] To get a cleaner equation, multiply the numerator and denominator by 3 to eliminate the fraction: \[ r = \frac{12}{3 + 4\cos \theta} \]

Step 6: Verify the Equation

Verify that the form of the equation meets the conditions given. The conic section is a hyperbola since the eccentricity \( e = \frac{4}{3} \) is correct and the focus is at the pole.

Key Concepts

Eccentricity

Eccentricity is a crucial concept when studying conic sections, as it determines the shape of the curve in the plane. Represented by the symbol \( e \), eccentricity helps to identify whether a conic is a circle, ellipse, parabola, or hyperbola. Understanding eccentricity can make it easier to predict and work with these shapes.

Here’s a simple way to remember:

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

In this exercise, the eccentricity \( e = \frac{4}{3} \), which means our conic section is a hyperbola. This is because its eccentricity is greater than one. Additionally, an increased value of eccentricity stretches the conic section further, making hyperbolas have two disconnected components, known as branches. This property affects how we write equations to accurately represent the conic in various coordinate systems.

Conic Sections

Conic sections are curves created as the intersection of a plane and a double-napped cone. They are a foundational element of geometry and include circles, ellipses, parabolas, and hyperbolas.

Each type of conic section has distinct properties and equations to represent them. Here’s a breakdown:

• **Circle**: Forms when the plane cuts perpendicular to the axis of the cone.
• **Ellipse**: Formed when the plane cuts at an angle, but does not pass through the base.
• **Parabola**: Occurs when the plane is parallel to a generatrix of the cone.
• **Hyperbola**: Results when the plane intersects both halves of the cone. It essentially forms two "U" shaped curves mirroring each other.

The type of conic is determined by the eccentricity and can be expressed in polar coordinates by aligning the focus with the pole. In polar coordinates, a conic's equation depends on its eccentricity and directrix, showcasing the direct relationship between these elements.

Hyperbolas

Hyperbolas are fascinating because they consist of two separate curves facing each other. This unique shape emerges when a cone is intersected by a plane at an angle steeper than the normal conic sections.

A hyperbola features two key properties:

• **Asymptotes**: Imaginary lines that the hyperbola approaches but never touches. They help in graphing and understanding the shape.
• **Vertices**: Points where the hyperbola touches an axis, marking the closest or farthest points of each branch from the center.

The polar equation of a hyperbola with focus at the pole can be derived using its eccentricity and the equation of its directrix. For a given directrix, the general form is \( r = \frac{ed}{1 + e \cos \theta} \). This equation describes the position of any point on the hyperbola in relation to the focus.

In the context of the exercise, using \( e = \frac{4}{3} \) and the directrix \( r \cos \theta = -3 \), the final polar form of the hyperbola becomes \( r = \frac{12}{3 + 4\cos \theta} \). This showcases the hyperbola with respect to its eccentricity and locates its directrix vertically, providing the geometric traits needed for understanding and plotting this curve.