Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Hyperbola} &e=2&x=1\end{array}$$

Step-by-Step Solution

Answer

The polar equation of the hyperbola is \(r = \frac{2}{1 - 2cos(\theta)}\).

Step 1: Identify the parameters of the conic

The eccentricity \(e\) is given as 2 and the equation of the directrix \(x = d\) implies that \(d = 1\).

Step 2: Apply the formula of the conic

Plug these values into the polar equation for a hyperbola. We use the negative version of the formula, since for a hyperbola the denominator of the formula is \(1 - ecos(\theta)\), unlike for an ellipse, where it's \(1 + ecos(\theta)\). Therefore, the polar equation of the conic is \(r = \frac{2(1)}{1 - 2cos(\theta)}\). We choose the version with the minus sign because we are dealing with a hyperbola, not an ellipse.

Step 3: Simplified form of the equation

You can simplify the polar equation to its final form: \(r = \frac{2}{1 - 2cos(\theta)}\).

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. These curves play an important role in geometry and can be classified into different types such as circles, ellipses, parabolas, and hyperbolas.

Each type of conic section is defined by its unique set of properties and equations. The overall family of conic sections represent different slices of a cone:

• Circle: The section made when the cutting plane is parallel to the base of the cone.
• Ellipse: Formed when a plane slices through a cone at an angle, cutting through both nappes.
• Parabola: Results when the plane is parallel to one of the slant heights of the cone.
• Hyperbola: Occurs when the plane cuts through both nappes of the cone without being parallel to the axis.

Understanding conic sections helps in various applications in physics, engineering, and computer graphics.

Hyperbola

A hyperbola is one of the conic sections and is realized when a plane cuts through both nappe of a cone in a way that produces two distinct, mirror-image curves. This curve features two branches that open either sidewards or up and down, characterized by two foci from which distances vary.

Key features of a hyperbola:

• Vertices: Points where the hyperbola intersects its axis.
• Foci: Points away from which distances are measured.
• Asymptotes: Imaginary lines that the arms of the hyperbola approach but never meet.

The hyperbola's equation depends on these foci and is structured differently compared to circles or parabolas. The polar equation for hyperbolas takes into consideration its eccentricity and orientation.

Eccentricity

Eccentricity is a parameter associated with every conic section and denotes how stretched or open the conic is. It takes a specific numeric value which indicates the conic's shape.

For conic sections:

• Circle: Eccentricity (\( e \)) = 0
• Ellipse: 0 < (\( e \)) < 1
• Parabola: Eccentricity (\( e \)) = 1
• Hyperbola: (\( e \)) > 1

The eccentricity of a hyperbola, in particular, is greater than 1, which signifies that it opens widely as compared to ellipses. This value, paired with the directrix, helps in formulating a conic's equation.

Polar Equation

A polar equation expresses a curve using polar coordinates, where each point on the plane is defined by a distance from a reference point (often called the pole) and an angle from a reference direction.

In the context of conic sections, specifically a hyperbola, the polar equation serves as an efficient means to describe the configuration in terms of its eccentricity and directrix.

The polar form of a hyperbola is expressed as \[ r = \frac{ed}{1 - e \cos(\theta)} \]where:

• \( r \): the radius or distance from the pole.
• \( e \): the eccentricity of the conic.
• \( d \): the distance to the directrix.
• \( \theta \): the angle from the baseline.

This formulation helps simplify many computations in problems involving conic sections.