Problem 54 - College Algebra

Question

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=2 ; e=2.5\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{5}{1 + 2.5 \sin \theta} \).

Step 1: Understanding the Problem

We need to find the polar equation of a conic with given eccentricity (\(e=2.5\)) and directrix (\(y=2\)). The focus of the conic is at the origin.

Step 2: Identifying the Type of Conic

Since the eccentricity \(e = 2.5\) is greater than 1, the conic is a hyperbola.

Step 3: Standard Form for Polar Equations of Conics

The general form of the polar equation for a conic is \(r = \frac{ed}{1 + e\sin\theta}\) if the directrix is horizontal (parallel to the x-axis), and \(r = \frac{ed}{1 + e \cos \theta}\) if the directrix is vertical.

Step 4: Converting the Directrix to Polar Form

Since the directrix is \(y = 2\), which is horizontal, we use the polar form equation \(r = \frac{ed}{1 + e\sin\theta}\). Here the distance \(d\) from the focus (origin) to the directrix is 2.

Step 5: Substituting Values into the Equation

Substitute \(e = 2.5\) and \(d = 2\) into the equation for a horizontal directrix: \[ r = \frac{2.5 \times 2}{1 + 2.5 \sin \theta} \]Simplifying, we get\[ r = \frac{5}{1 + 2.5 \sin \theta} \].

Step 6: Final Polar Equation of the Conic

The polar equation of the conic with the given parameters is \[ r = \frac{5}{1 + 2.5 \sin \theta} \].

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each type is determined by the angle at which the plane intersects the cone and the relative proximity of the intersecting plane to the cone's vertex. For example:

• A circle is formed when the plane intersects the cone parallel to its base.
• An ellipse is a stretched circle, formed when the intersection plane is tilted but does not reach the base.
• A parabola emerges when the plane is parallel to the side of the cone.
• A hyperbola occurs when the plane intersects both halves of the double cone.

Conic sections have profound applications in fields like astronomy, physics, and engineering. Understanding their properties can help solve real-world problems, such as calculating orbits or designing lenses.

Eccentricity

Eccentricity (denoted as \(e\)) measures how much a conic section deviates from being circular. It helps in determining the shape and type of conic sections. Here's how eccentricity defines the conic:

• For a circle, \(e = 0\).
• If \(0 < e < 1\), the conic is an ellipse.
• When \(e = 1\), it forms a parabola.
• If \(e > 1\), the conic is a hyperbola.

In the given problem, the eccentricity is \(2.5\), indicating a hyperbola. Higher eccentricity signifies more elongation from a perfect circle, thus providing insights into the conic's nature and structure. Understanding eccentricity helps in visualizing and analyzing conics analytically in mathematical space and practically in phenomena such as planetary orbits.

Directrix

A directrix in conic sections is a fixed line used in describing and defining the curve. It differs depending on the conic:

• In ellipses and parabolas, it serves as a reference line which helps in determining how points on the conic relate to the focus.
• For hyperbolas, like the one in the problem, it is pivotal as it helps determine the distance measurement crucial for the equation in polar form.

In the given exercise, the directrix is a horizontal line at \(y = 2\). This line assists in creating the polar equation of the hyperbola. The relationship between the point on the conic, its focus, and the directrix is what defines the properties and the equation of the conic. A well-understood directrix helps formulate its equation efficiently.

Focus

The focus is a point that plays a key role in defining a conic section. Every point on the conic is equidistant from the focus and another geometric feature (such as the directrix). Here's its role in different conics:

• For a circle and an ellipse, there are one or more foci which affect the shape.
• A parabola has one focus, which with the directrix, defines its characteristic curve.
• A hyperbola, like in this exercise, has two foci but only one is usually discussed in simple polar equations.

In this scenario, the focus is at the origin (0,0), serving as the central fixed point from which distances to the directrix and points on the hyperbola are measured. Understanding the role of the focus is vital in forming precise equations and interpreting the geometry of conics.