Problem 44 - Contemporary Precalculus

Question

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Eccentricity \(4 / 5 ;\) directrix: \(r=3 \csc \theta\)

Step-by-Step Solution

Answer

Answer: The polar equation of the conic section is \(r = \frac{12}{5 + 4 \cos \theta}\).

Step 1: Find the distance d

To find d, use the equation of the directrix \(r = 3 \csc \theta\). When \(\theta = 90^\circ\), \(\csc \theta = 1\), and the equation becomes \(r = 3\). So, the distance from the pole (0,0) to the directrix is 3 units.

Step 2: Plug in the given values into the polar equation formula

Now that we know the distance d = 3 and the eccentricity e = \(\frac{4}{5}\), plug these values into the polar equation formula: \(r = \frac{ed}{1 + e \cos \theta} = \frac{(\frac{4}{5})(3)}{1 + (\frac{4}{5}) \cos \theta}\)

Step 3: Simplify the equation

Simplify the polar equation by multiplying the numerator and denominator terms: \(r = \frac{12}{5} \cdot \frac{1}{1 + \frac{4}{5} \cos \theta} = \frac{12}{5 + 4 \cos \theta}\)

Step 4: Final polar equation

The polar equation of the conic section with focus at (0,0), eccentricity \(\frac{4}{5}\), and directrix \(r = 3 \csc \theta\) is given by: \(r = \frac{12}{5 + 4 \cos \theta}\)

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. Depending on the angle and position of the plane, the intersection can form different types of curves such as circles, ellipses, parabolas, or hyperbolas. These curves are fundamental in mathematics and arise naturally in various physical situations, like orbits of planets or reflections of light.
A circle is a special kind of ellipse where the focal points coincide, whereas an ellipse has distinct foci, making it slightly stretched in one or multiple directions. Parabolas are unique because they have a single focal point and a direct line, known as the directrix, that they "mirror off of," making them excellent for some optical devices. Hyperbolas have two separate curves, each corresponding to a focus and directrix.

• The type of conic is determined by eccentricity: for circles (\(e = 0\)), ellipses (\(0 < e < 1\)), parabolas (\(e = 1\)), and hyperbolas (\(e > 1\)).

Eccentricity

Eccentricity is a number that describes how much a conic section deviates from being circular. For any conic section, the eccentricity (denoted as \(e\)) is defined as the ratio of the distance from any point on the conic to its focus, to the perpendicular distance from the point to the directrix.
In simple terms, eccentricity helps us understand the shape of the curve.

• If \(e = 0\), the conic is a perfect circle.
• For values \(0 < e < 1\), the conic is an ellipse, and the curve will be stretched more perpendicularly. The small eccentricity makes the ellipse more circular.
• At \(e = 1\), the conic becomes a parabola, an important shape in optics and engineering.
• If \(e > 1\), the conic is a hyperbola, with two separate branches.

Using the eccentricity value of \(\frac{4}{5}\), we can determine that the conic is an ellipse because this value is less than 1.

Directrix

The directrix of a conic section is a line used in the conic's definition along with a point called the focus. For every point on the conic, the ratio of its distance to the focus and to the directrix is constant and equal to the eccentricity. This line plays a vital role in defining the shape of the conic section:
The directrix helps maintain a consistent relationship between the focus and the curvature of the conic. By keeping the ratio with the eccentricity

• The line's distance from the origin affects the stretch and orientation of the curve.
• For ellipses and hyperbolas, directrices help form the bounds for the curves.
• For parabolas, the directrix is one of the guiding lines parallel to which the cone is sliced in order to form the shape.

In this exercise, the directrix given is \(r = 3 \csc \theta\), indicating that as theta reaches \(90^\circ\), the distance from the pole to the directrix is exactly 3.

Polar Coordinates

Polar coordinates provide a way to represent points in a plane using the distance from a reference point and an angle from a reference direction. This system is particularly useful when dealing with curves that are radially symmetric, like many conic sections.

• In the polar coordinate system, a point is represented by \((r, \theta)\), where \(r\) is the radial distance from the origin (pole), and \(\theta\) is the angle measured counterclockwise from the positive x-axis.

Polar equations, such as \(r = \frac{12}{5 + 4 \cos \theta}\), describe the location and shape of curves directly in terms of \(r\) and \(\theta\). This method can simplify equations compared to Cartesian coordinates, especially when dealing with curves like circles and spirals. In this exercise, polar coordinates are used to describe the conic sections, making it easy to visualize how changes in \(\theta\) shift points around the origin.