Problem 36 - Thomas Calculus

Question

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=1 / 3, \quad y=6$$

Step-by-Step Solution

Answer

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

Step 1: Identify the conic type

The eccentricity \( e = \frac{1}{3} \) is less than 1, indicating that the conic section is an ellipse.

Step 2: Express the directrix in polar coordinates

The directrix given is \( y = 6 \). This is a horizontal line passing through \( (0, 6) \). In polar coordinates, where \( r \cos \theta = x \) and \( r \sin \theta = y \), the directrix can be written as \( r \sin \theta = 6 \).

Step 3: Write the polar equation of the ellipse

The polar equation for a conic section with eccentricity \( e \) and directrix \( r \sin \theta = d \) is given by: \[ r = \frac{ed}{1 + e \sin \theta} \]. Substitute \( e = \frac{1}{3} \) and \( d = 6 \) into the formula: \[ r = \frac{\left(\frac{1}{3}\right) \cdot 6}{1 + \frac{1}{3} \sin \theta} = \frac{2}{1 + \frac{1}{3} \sin \theta} \].

Step 4: Simplify the polar equation

Simplify the fraction: \[ r = \frac{2}{1 + \frac{1}{3} \sin \theta} = \frac{2}{\frac{3 + \sin \theta}{3}} = \frac{6}{3 + \sin \theta} \]. So the polar equation of the ellipse is: \[ r = \frac{6}{3 + \sin \theta} \].

Key Concepts

Understanding Ellipses

An ellipse is a fascinating conic section that resembles an elongated circle. It is defined as the set of points for which the sum of the distances to two fixed points, known as foci, is constant. An ellipse has several important features: the major and minor axes and its wide-ranging applications, from planetary orbits to architectural designs. At its core, an ellipse is symmetric about its axes, with its center being the midpoint between its foci.

When dealing with ellipses in mathematical exercises, particularly in polar coordinates, one defining property to look at is their eccentricity. This value, when less than 1, confirms that the conic section in question is indeed an ellipse.

• Symmetry: Is symmetrical around its two axes.
• Eccentricity: Always less than 1.
• Special Case: Becomes a circle when eccentricity is 0.

Polar Coordinates: A Unique Perspective

Polar coordinates provide a unique method of graphing that extends beyond traditional Cartesian coordinates. In this system, each point is represented by a radius and angle from a central point (usually the origin). Polar coordinates are especially useful for describing curves and shapes that involve rotations, such as spirals and conics, including ellipses.

For an ellipse, the polar equation might appear slightly complex, but it fundamentally involves substituting known values like eccentricity and the directrix into a standard equation. Knowledge of a directrix makes conversion and representation in polar form simpler and more intuitive.

• Components: Defined by a radius and an angle. Useful for curves and rotational shapes.
• Conversion: Converts conic sections such as ellipses into simpler equations.

Understanding these coordinates helps in observing the intricate behavior of ellipses as they rotate around different axes.

Eccentricity: Defining the Shape

Eccentricity in geometry is a key value that helps define the shape of a conic section. It measures how much a conic deviates from being circular. For ellipses, the eccentricity value is between 0 and 1, providing a clear indication that the curve is an elongated circle rather than a perfect one.

An eccentricity of 0 equals a perfect circle, where both foci are at the same point. As the eccentricity increases towards 1, the ellipse becomes more stretched. Importantly, this value then goes on to affect the formulas we use when translating into polar coordinates. In the problem, since the eccentricity is given as 1/3, it reinforces that the conic section is an ellipse.

• Value: Lies between 0 and 1 for ellipses.
• Influence: Determines how circular or elongated the ellipse will be.
• Role in Equations: Critical for setting up polar equations.

Thus, by understanding an ellipse's eccentricity, one can predict its appearance and behavior in a polar coordinate system.