Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r(4-5 \sin \theta)=1\)

Step-by-Step Solution

Answer

The conic is a hyperbola with eccentricity 5 and directrix \( y = \frac{1}{5} \).

Step 1: Simplify the given equation

The given polar equation is \( r(4 - 5 \sin \theta) = 1 \). We start by solving for \( r \):\[r = \frac{1}{4 - 5 \sin \theta}\]

Step 2: Identify the form of conic section

The equation \( r = \frac{1}{4 - 5 \sin \theta} \) is of the form \( r = \frac{ed}{1 - e \sin \theta} \). Compare this with \( r = \frac{ed}{1 + e \sin \theta} \) to identify the type of conic:- Here, \( e = 5 \) and \( ed = 1 \). The expression is \( ed = 1 \), hence \( d = \frac{1}{5} \).

Step 3: Determine eccentricity (e)

From Step 2, we noted that \( e = 5 \). Typically, if \( e > 1 \), the conic section is a hyperbola.

Step 4: Identify the directrix

The expression \( ed = 1 \) allows us to compute the directrix \( d \):\[d = \frac{1}{e} = \frac{1}{5}\]Since the expression is \( 1 - e \sin \theta \), the directrix is horizontal (parallel to the \( x \)-axis).

Key Concepts

Polar Coordinates

In mathematics, polar coordinates are a way of representing points in a plane using an angle and a distance. Unlike Cartesian coordinates, which use a grid with x and y values, polar coordinates describe each point by its distance from a fixed point known as the origin and an angle measured from a fixed direction. The distance to the point is usually denoted as \(r\) and the angle as \(\theta\).

• The point \((r, \theta)\) means you're \(r\) units away from the origin.
• \(\theta\) is the angle formed clockwise from the positive x-axis.

Polar coordinates are particularly useful when dealing with problems involving curves that wrap around a point, like circles or spirals. In the exercise above, the given equation \( r(4 - 5 \sin \theta) = 1 \) is written in polar form.

Hyperbola

A hyperbola is one of the types of conic sections, which are curves obtained by intersecting a cone with a plane at different angles. A hyperbola consists of two separate curves or branches that mirror each other. It's characterized by its degree of openness, which is influenced by its eccentricity.

• For hyperbolas, the equation involves reflecting points across a certain direction.
• In polar form, hyperbolas are recognizable if the eccentricity \(e\) is greater than 1.

In the solution provided above, the polar equation \( r = \frac{1}{4 - 5 \sin \theta} \) was identified to represent a hyperbola. This determination was made based on its eccentricity \(e = 5\), which is greater than 1, thus fulfilling the characteristic condition of a hyperbola.

Eccentricity

Eccentricity is a number that describes the shape of a conic section. It denotes how much the conic section deviates from being circular. For any conic, the eccentricity \(e\) provides a specific clue:

• When \(e = 0\), the conic is a circle.
• When \(0 < e < 1\), the conic is an ellipse.
• When \(e = 1\), the conic is a parabola.
• When \(e > 1\), as in our problem, the conic is a hyperbola.

In the exercise solution, it was identified that the hyperbola had an eccentricity of 5. This large value confirms the hyperbola’s wide openness since, theoretically, as the value of \(e\) grows, the branches of the hyperbola open outward more.

Directrix

The directrix of a conic section is a line associated with each conic, providing a reference to determine how far or close a point on the conic is, compared to the conical's focus. It helps further describe the degree and direction of the curve with respect to the coordinate axes.

• For hyperbolas and ellipses, each has two directrices, and they help define the curve.
• In the polar form equation present in our exercise, the element "d" represents the distance to the directrix.

In the original problem's solution, the directrix was calculated using the relationship \( ed = 1 \). With \(e = 5\), the computed directrix \(d = \frac{1}{5}\) indicates its horizontal positioning, parallel to the x-axis, serving as a crucial guide for drawing the conic section.