Problem 17 - Precalculus with Calculus Previews

Question

Find a polar equation of the conic with focus at the origin that satisfies the given conditions. $$ e=1, \operatorname{directrix} x=3 $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{3}{1 + \cos(\theta)} \).

Step 1: Identify the Type of Conic

Since the eccentricity, denoted as \( e \), is equal to 1, the conic is a parabola.

Step 2: Understand the Directrix

The given directrix is a vertical line, \( x = 3 \). Since the directrix is vertical and the focus is at the origin, the axis of symmetry for the parabola is horizontal.

Step 3: Choose the Correct Polar Form

A polar equation for a conic with a vertical directrix has the form \( r = \frac{ed}{1 + e\cos(\theta)} \) when the directrix is \( x = d \) and the focus is at the origin.

Step 4: Substitute Values into the Polar Equation

Here, \( e = 1 \) and \( d = 3 \). Substitute these values into the polar form: \( r = \frac{1 \cdot 3}{1 + 1\cos(\theta)} \).

Step 5: Simplify the Expression

Simplify the equation to get the final polar equation of the conic: \( r = \frac{3}{1 + \cos(\theta)} \).

Key Concepts

Eccentricity

In the context of conic sections, eccentricity plays a crucial role in defining the shape of the conic. Eccentricity, often denoted by the letter "e", is a number that represents the degree of deviation of the conic section from being a perfect circle. A circle has an eccentricity of zero.
A conic section can be an ellipse, parabola, or hyperbola, determined by its eccentricity value. When eccentricity (\( e \)) equals 1, the conic is a parabola. An important property of parabolas and eccentricity is that parabolas have the geometric property of being equidistant from a point (focus) and a line (directrix).

• If \( e < 1\), the conic is an ellipse, indicating it is closer to circular.
• If \( e = 1\), it is a parabola, reflecting an equal distance relationship to the focus and directrix.
• If \( e > 1\), the conic is a hyperbola, illustrating a more open, diverging path.

Understand that eccentricity is fundamental in identifying the type of conic and helps in constructing its equation.

Parabolas

A parabola is a unique type of conic section that is perfectly symmetrical. It can appear open either vertically or horizontally depending on the position of its directrix and the focus. When given in polar form, parabolas exhibit this symmetry as well.
In any case, the axis of symmetry will always be perpendicular to the directrix. In polar coordinates, the equation of a parabola with the focus at the origin can appear as \( r = \frac{ed}{1+e\cos(\theta)}\) or \( r = \frac{ed}{1+e\sin(\theta)}\). The form depends on whether the directrix is vertical or horizontal:

• For vertical directrices, the equation involves \( \cos \theta \).
• For horizontal directrices, the equation involves \( \sin \theta \).

This equation provides the relationship between the radial distance, \( r \), and the angle \( \theta \) from the polar axis. Because \( e \) is 1 in the context of parabolas, solving these equations provides a straightforward path to their geometric representation.

Directrix in Conic Sections

The directrix is a significant, guiding line used in the definition of all conic sections, including parabolas. In the case of a parabola, it plays a vital role in determining the shape and orientation of the curve.
The directrix for a parabola with focus at the origin is a line external to the parabola that maintains a consistent distance relationship to it.Imagine a parabola where every point is equidistant from a point (the focus) and a line (the directrix). This equidistance is a definitive characteristic of parabolas.

• In the exercise provided, the directrix is a vertical line, \( x = 3 \).
• That placement means the parabola opens horizontally along the x-axis, as its focus is at the origin.

Understanding the position of the directrix helps not only in forming the polar equations but also in visualizing the parabola's orientation and essential symmetry.