Problem 21 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{ed}{1 - e \cos \theta}\).

Step 1: Understand the Parameters of a Conic

The parameters given for the conic are:- Focus at the origin- Eccentricity \(e\)- Directrix \(x = -d\)Using these, the task is to find a polar equation that relates these properties.

Step 2: Recall the Definition of a Conic

A conic section is defined as the locus of points whose distance from a focus is a constant multiple (eccentricity \(e\)) of the distance from a directrix. The polar form of a conic can be expressed as \(r = \frac{ed}{1 - e \cos \theta}\) where \(d\) is the distance from the focus to the directrix.

Step 3: Set Up Cartesian and Polar Relationships

The directrix \(x = -d\) in Cartesian form needs to relate to \(r\) and \(\theta\) in polar form. The general relationship for a conic in polar coordinates is known as \(r = \frac{ed}{1 - e \cos \theta}\). Let's verify the directrix' role.

Step 4: Substitute and Simplify

To maintain the directrix condition, consider a point \((x,y)\) that translates in polar as \(x = r \cos \theta\) and \(y = r \sin \theta\). Given the directrix \(r \cos \theta = -d\), substitute into the conic section equation:Subtract \(e \cos \theta\) from both sides:\(r (1 - e \cos \theta) = ed\) Now, solve for \(r\):\[r = \frac{ed}{1 - e \cos \theta}\]

Step 5: Final Verification

Ensure all terms and conditions have been used:- Origin as focus.- Conforms to directrix \(x = -d\).- Eccentricity applied in polar coordinates by deduction.The derived formula is consistent with the polar description of a conic section.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. Depending on the angle and location of the intersection, we get different types of conics: ellipses, parabolas, and hyperbolas.
Each conic section has unique properties and a different eccentricity that defines its shape.

• Ellipse: Eccentricity is less than 1. The set of all points where the sum of distances from two fixed points (foci) is constant.
• Parabola: Eccentricity equals 1. The set of all points that are equidistant from a single focus and a directrix.
• Hyperbola: Eccentricity is greater than 1. Defined as the set of all points where the difference of distances from two foci is constant.

Understanding how conic sections behave and relate to circles (a special case with eccentricity 0) helps us in fields like astronomy and engineering.
In polar coordinates, these shapes can be expressed with specific equations that define their curves.

Eccentricity

Eccentricity is a parameter that determines the type and shape of a conic section. It is denoted by the letter \(e\). Essentially, eccentricity describes how "stretched" a conic is compared to a circle.

• Ellipse: 0 < \(e\) < 1
• Parabola: \(e\) = 1
• Hyperbola: \(e\) > 1

This simple number gives us so much information! In polar coordinates, it determines how far the focus of the conic is from its center or directrix.
Higher eccentricities stretch the conic more, which is why hyperbolas look more open compared to ellipses and parabolas.

Directrix

The directrix is a fixed line used in describing a conic section. When combined with a focus, it helps define the conic type. The distance between a point on the conic and the directrix plays into the equation with the eccentricity.

• For an ellipse and hyperbola, the directrix serves as a measure of the "outward" stretch of the curve.
• For parabolas, it is used together with the focus to give that "correct" shape.

In the polar equation \(r = \frac{ed}{1 - e \cos \theta}\), the term \(d\) represents the perpendicular distance from the pole (the origin) to the directrix.
This relationship is crucial in deriving the polar form of the conic equation.

Polar Equation

A polar equation describes a conic section in terms of radial distance \(r\) and angle \(\theta\). This method is beneficial in situations where symmetry around a point makes Cartesian coordinates less effective.

• Polar equations can easily express circular or spiral shapes.
• For conic sections, it's represented as \(r = \frac{ed}{1 - e \cos \theta}\), where \(e\) is the eccentricity and \(d\) is the distance from the pole to the directrix.

Polar coordinates are especially important in areas dealing with orbits, like astronomy, since the planets revolve around the sun in elliptical orbits.
By setting up the equation, you can directly relate the geometric property of a conic (eccentricity and directrix) to its polar form, allowing for a clearer understanding and manipulation of these curves.