Problem 23 - Calculus Early Transcendentals

Question

Show that a conic with focus at the origin, eccentricity \(e,\) and directrix \(y=-d\) has polar equation $$r=\frac{e d}{1-e \sin \theta}$$

Step-by-Step Solution

Answer

The conic's polar equation is \( r = \frac{ed}{1 - e \sin \theta} \).

Step 1: Define the properties of the conic

For any conic section, the definition involving the eccentricity and directrix is given by the ratio of the distance of a point from a focus to its distance from a directrix. The generic definition is: \( e = \frac{r}{p} \), where \( r \) is the distance from the focus, and \( p \) is the perpendicular distance from the directrix to the point on the conic.

Step 2: Set up the relation using polar coordinates

In polar coordinates, let the directrix be described as \( y = -d \). A point \((r, \theta)\) has a distance \( r \) from the focus, and from the definition of the conic related to the directrix, we use the perpendicular distance formula for a point \((r,\theta)\) to directrix \(y = -d\), given by \( |r \sin \theta + d| \).

Step 3: Relate distance from focus and directrix

According to the definition of conic sections where the focus is at the origin, \( e = \frac{r}{|r \sin \theta + d|} \). Simplify this to the absolute distance: \( er \sin \theta + ed = r \).

Step 4: Solve for \( r \)

Re-arrange the equation \( er \sin \theta + ed = r \) to solve for \( r \). This gives: \( r - er \sin \theta = ed \), and then \( r(1 - e \sin \theta) = ed \). Thus, \( r = \frac{ed}{1-e \sin \theta} \).

Step 5: Confirm correctness of polar equation

We have deduced that the polar equation of the given conic section is \( r = \frac{ed}{1-e \sin \theta} \). The calculation is derived through the conic definition involving eccentricity and the directrix, verifying the initial statement.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to represent points in a plane using distance and angle. Instead of using the regular Cartesian coordinates (x, y), the polar system describes locations using a radius, denoted as \( r \), and an angle, \( \theta \), which is measured from the positive x-axis.

• The radius \( r \) represents the distance from the origin to a point.
• The angle \( \theta \) identifies the direction from the origin to the point.

To convert between polar and Cartesian coordinates:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

Polar coordinates can simplify the equations of curves, especially when dealing with circular or spiral shapes. In the context of conic sections such as parabolas, ellipses, and hyperbolas, polar coordinates can reveal properties more elegantly than the Cartesian format.
They become particularly useful when the conic's focus is at the origin, allowing direct relations involving angles and distances.

Eccentricity

Eccentricity \( e \) is a fundamental property defining the shape of a conic section. It measures the deviation of the conic from being circular:

• For a circle, \( e = 0 \).
• For an ellipse, \( 0 < e < 1 \).
• For a parabola, \( e = 1 \).
• For a hyperbola, \( e > 1 \).

Eccentricity determines how 'stretched' a conic is. It is calculated by comparing the distance of any point on the conic from a fixed point (focus) to the perpendicular distance from a fixed line (directrix), maintained as a constant ratio at every point:\[ e = \frac{r}{|r \sin \theta + d|} \]
In the given exercise, adjusting the value of \( e \) changes the balance between the pull towards the focus and the positioning relative to the directrix.

Directrix

A directrix is a fixed line used in the geometric definition of a conic section. Each conic section type has its own orientation and position for its directrix:

• For ellipses and hyperbolas, there are typically two directrices.
• For parabolas, there is one.

The distance of any point on a conic section from the directrix helps describe how the conic "focuses" or converges around the focus.
In polar coordinates, if a directrix is described by an equation like \( y = -d \), it serves as a benchmark measuring the perpendicular distance from a point on the conic to this line. This relationship is utilized directly in calculating eccentricity and establishing the polar form equation.
In our context, the directrix aligns perpendicularly to the conic's axis and helps define the conic alongside its focus and eccentricity.

Focus of a Conic

The focus of a conic section is a crucial point that lies at the center of interest for determining the conic's shape. It's the anchor:

• For ellipses and hyperbolas, there are two foci.
• For circles, the center acts as the single focus.
• Parabolas have one focus.

In the case of the given problem, the focus is at the origin of the polar coordinate system, which simplifies a lot of calculations.
The polar equation of conics leverages this position by expressing relationships relative directly to this focus, such as measuring the distance \( r \) of a point from the focus. The fixed nature of the focus at the origin facilitates the straightforward manipulation and simplification of conic equations into the elegant polar form \( r = \frac{ed}{1-e \sin \theta} \).

Polar Equation

A polar equation describes a curve as a function of angles and distances, expressed in polar coordinates. For conic sections, it provides a powerful way to investigate deeper relationships between the geometry and algebra:

• The polar form of a conic with a focus at the origin simplifies complex algebraic expressions.
• The derived equation \( r = \frac{ed}{1-e \sin \theta} \) captures the relationship involving eccentricity \( e \), focus, and directrix directly.

This form stems from the definition of eccentricity and directrix, connecting the distance from focus to a generic point on the conic and its relative distance to the directrix.
Using polar equations, one can easily sketch and analyze conic curves by varying eccentricity, changing from ellipses to parabolas, to hyperbolas, reflecting how the conic's shape transitions as \( e \) evolves.