Problem 18 - Precalculus with Calculus Previews

Question

Find a polar equation of the conic with focus at the origin that satisfies the given conditions. $$ e=\frac{3}{2}, \text { directrix } y=2 $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{6}{2 + 3 \sin \theta} \).

Step 1: Identifying the type of conic

The eccentricity is given as \(e = \frac{3}{2}\). Since \(e > 1\), the conic is a hyperbola.

Step 2: General polar equation for a conic

The general form of a polar equation with focus at the origin is given by \(r = \frac{ed}{1 + e \sin \theta}\) or \(r = \frac{ed}{1 + e \cos \theta}\), depending on the orientation of the directrix.

Step 3: Deciding polar equation form

Since the directrix is vertical \(y = 2\) and parallel to the polar axis, use \(r = \frac{ed}{1 + e \sin \theta}\) where the directrix is in the form \(y = k\).

Step 4: Finding directrix distance (d)

The directrix \(y = 2\) implies that the perpendicular distance from the origin to the line is \(d = 2\).

Step 5: Substituting known values

Substitute \(e = \frac{3}{2}\) and \(d = 2\) into the polar equation: \[ r = \frac{\frac{3}{2} \times 2}{1 + \frac{3}{2} \sin \theta} \] which simplifies to \[ r = \frac{3}{1 + \frac{3}{2} \sin \theta} \].

Step 6: Simplifying the equation

Multiply the numerator and the denominator by 2 to eliminate the fraction in the denominator, resulting in \[ r = \frac{6}{2 + 3 \sin \theta} \].

Key Concepts

Eccentricity

Eccentricity is a key concept when discussing conic sections, as it determines the shape of a conic. In the given exercise, the eccentricity is specified as \( e = \frac{3}{2} \). In general, eccentricity (\( e \)) tells us how much a conic deviates from being circular.

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

Given that \( e = \frac{3}{2} \) in our problem, we know the conic is a hyperbola. Hyperbolas have an eccentricity greater than 1, indicating two mirror-symmetric curves which open in opposite directions. Understanding eccentricity helps in identifying and studying the behavior of various conics.

Hyperbola

A hyperbola is a significant type of conic section that appears when the eccentricity \( e \) is greater than 1, as identified in this exercise. It represents two sets of divergent curves.

• The two branches of a hyperbola are mirror images of each other.
• They extend to infinity and are defined by their asymptotes.
• The hyperbola has reflective symmetry around its center, which lies exactly midway between its branches.

Hyperbolas have unique equations depending on their orientation and reference axes. The polar equation for a hyperbola is typically of the form \( r = \frac{ed}{1 \pm e \sin \theta} \) or \( r = \frac{ed}{1 \pm e \cos \theta} \), depending on whether the directrix is vertical or horizontal. In our case, given that the directrix is vertical, we use a form involving \( \sin \theta \). Hyperbolas are critical in fields like physics and astronomy, especially when orbit paths and waves are analyzed.

Polar Equation

Polar equations are a way to express conics using polar coordinates, which consider distance from a central point (the pole) and angles from a reference direction. The polar equation used here is necessary because the focus of the conic is at the origin.

• This approach simplifies calculations for conics oriented around a point.
• The form used depends on the orientation of the directrix relative to the polar axis, which can be horizontal or vertical.

For the current exercise, the directrix is \( y = 2 \), which means it is a vertical line parallel to the polar axis. Thus, the equation for the hyperbola takes the form \( r = \frac{ed}{1 + e \sin \theta} \), where \( d \) is the distance from the origin to the directrix.
After plugging in \( e = \frac{3}{2} \) and \( d = 2 \) into this expression, it's simplified to \( r = \frac{6}{2 + 3 \sin \theta} \). Polar equations offer a streamlined method for graphing and analyzing conics in polar coordinates, especially useful in recognizing patterns and directional dependencies.