Problem 45 - College Algebra

Question

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

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{4}{1 - 2 \sin \theta} \).

Step 1: Understanding the Problem

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

Step 2: Choosing the Polar Equation Form

For a conic with the focus at the origin, the general polar equation is \(r = \frac{ed}{1 - e \sin \theta}\) or \(r = \frac{ed}{1 + e \sin \theta}\), depending on the position of the directrix (vertical or horizontal). Here, as the directrix is horizontal (\(y = 2\)), the equation will be based on \(\sin \theta\).

Step 3: Substitute Eccentricity and Directrix Distance

Since the directrix is \(y = 2\), the distance \(d\) from the origin to the directrix is 2. The eccentricity \(e\) given is 2. Substitute these values into the polar equation: \[ r = \frac{2 \times 2}{1 - 2 \sin \theta} \].

Step 4: Simplifying the Equation

Perform the multiplication in the numerator: \( r = \frac{4}{1 - 2 \sin \theta} \). This is the polar equation of the conic.

Key Concepts

Eccentricity

Eccentricity is a fundamental characteristic of conic sections, such as ellipses, parabolas, and hyperbolas. It is a numerical value that describes the shape and the nature of the conic. Here’s what you need to know about it:

• If the eccentricity (\(e\)) is less than 1, the conic is an ellipse.
• If \(e\) equals 1, the conic is a parabola.
• If \(e\) is greater than 1, as in our case (\(e = 2\)), the conic is a hyperbola.

In the context of our exercise, knowing that \(e = 2\) indicates we are dealing with a hyperbola. Eccentricity also helps in understanding how "stretched" a conic is. A higher eccentricity means the conic deviates more from being a perfect circle.
The formula for eccentricity varies depending on the conic. Still, its primary role is to relate the focus, directrix, and shape of the conic for accurate geometric representation.

Directrix

The directrix is a vital element used to define conic sections. It is a fixed line that, together with a point known as the focus, helps in constructing a conic section according to its eccentricity. Here’s a deeper look:

• A directrix serves as a reference line, to which the conic section’s distance is measured and compared with the distance to its focus.
• In mathematical terms, a conic section can be considered the locus of points whose distances to the focus and directrix are in a constant ratio, given by eccentricity \(e\).
• In our exercise, the directrix is the line \(y = 2\), which is horizontal.

This means any point on the conic will have its distance from the focus at the origin twice (since \(e = 2\)) the distance to the directrix. Understanding the role of the directrix is crucial in forming and interpreting the polar equation of a conic.

Polar coordinates

Polar coordinates are an alternative to the Cartesian coordinate system, ideal for objects and graphs involving circles and symmetry around a point. Let’s break it down:

• Instead of using \(x, y\) positions, polar coordinates use the radius \(r\) and the angle \(\theta\) from a specific axis.
• The polar coordinate system is particularly useful for describing the position of points along curves that have circular symmetry.
• The focus at the origin in our given problem also implies the centrality in polar coordinates, where everything is measured from this point.

The polar equation, \(r = \frac{ed}{1 - e \sin \theta}\), translates the spatial relationship defined by the eccentricity and directrix into this coordinate system. It makes it easier to plot and understand curved conic shapes in scenarios like ours, where the focus is at the origin and the directrix is given as horizontal.