Problem 42 - Precalculus

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc}\text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Ellipse} & e=\frac{3}{4} & y=-2 \end{array}$$

Step-by-Step Solution

Answer

The polar equation of the elliptical conic with a focus at the pole is \(r = \frac{8}{7 - 3* cos(\theta)}.\)

Step 1: Understand the formula for the conic section with a focus at the pole

The general formula for a conic section with a focus at the pole is \(r = \frac{p}{1 - e \cdot cos(\theta)}\) where \(r\) is the radius, \(p\) is the distance from the focus to the directrix, \(e\) is the eccentricity, and \(\theta\) is the angular coordinate.

Step 2: Calculate the value of p

The formula for \(p\) is \(p = \frac {d}{1+e}\), where \(d\) represents the distance from the pole to the directrix. The directrix is \(y=-2\), and the pole (focus) sits at the origin \((0, 0)\). The distance \(d\) is 2 and using the given eccentricity, \(p\) evaluates to \(p=\frac{2}{1+\frac{3}{4}} = \frac{8}{7}\).

Step 3: Substitute the values of e and p to get the polar equation

Substituting the values of \(p = \frac{8}{7}\) and \(e = \frac{3}{4}\) in the general formula, we get \(r = \frac{\frac{8}{7}}{1 - \frac{3}{4} \cdot cos(\theta)}.\) Simplifying, we get \(r = \frac{8}{7 - 3* cos(\theta)}.\)

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone at various angles. These shapes include circles, ellipses, parabolas, and hyperbolas. Each type arises from changing the angle at which you slice through.

• A circle forms when the slice is perfectly perpendicular to the cone's axis.
• An ellipse results from slicing at a shallow angle, while a parabola forms when the slice is parallel to one of the cone’s slopes.
• A hyperbola comes from intersecting the cone with a steeper angle, cutting through both halves of the cone.

Conic sections are essential in mathematics because they describe a lot of natural phenomena and engineering problems.
They can be expressed in Cartesian coordinates, but in polar coordinates -- like the exercise -- they are particularly useful when dealing with problems where symmetry around a point (the pole) is apparent.

Eccentricity

Eccentricity is a measure that describes how stretched out a conic section is. It is denoted by the symbol \( e \).

• For circles, \( e = 0 \). This means they have no stretch.
• Ellipses have eccentricity values between 0 and 1, \( 0 < e < 1 \), meaning they are slightly elongated.
• For parabolas, \( e = 1 \), indicating a perfect open curve.
• Hyperbolas have \( e > 1 \), showing more pronounced stretching.

The eccentricity value directly affects the shape and equation of a conic section when expressed in polar form.
In the exercise you are considering, the ellipse has an eccentricity of \( e = \frac{3}{4} \). This tells us it is an elongated circle, but does not have an open end like a hyperbola.

Directrix

A directrix is a fixed line used to help define and construct conic sections. This line maintains a constant relationship with all points on the conic.
In polar coordinates, a directrix serves as a reference line. Each point on the conic section is maintained at a specific location relative to this line.

• For an ellipse, like the exercise problem, the directrix helps define the elongation and orientation.
• Mathematically, it ties into the eccentricity and helps compute other parameters like \( p \), the semi-latus rectum.

The given directrix \( y = -2 \) means it is a horizontal line, two units below the pole -- our central focus point for the conic's polar equation.
The distance \( d = 2 \) from the pole to this directrix was used in solving the exercise.

Ellipse

An ellipse is a type of conic section characterized by its oval shape. Unlike a circle, its two focal points aren't equidistant from the center, causing it to elongate.
Ellipses can range from nearly circular to highly flattened shapes, depending on their eccentricity.
In the exercise, the polar equation derived describes such a shape:
\[ r = \frac{8}{7 - 3 \cdot \cos(\theta)} \]This equation reflects an ellipse with specific conditions:

• An eccentricity of \( \frac{3}{4} \), indicating moderate elongation.
• A directrix located horizontally at \( y = -2 \).

Polar equations are particularly vital in cases where one focus of the ellipse is at a significant position, such as the pole of the coordinate system. This can be handy in physics to model orbits of planets which are elliptical in nature.