Problem 34 - 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. $$e=1 / 4, \quad x=-2$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{2}{4 + \cos \theta} \).

Step 1: Understand Conic Section Parameters

We are given the eccentricity \(e = \frac{1}{4}\) and the directrix \(x = -2\). Since \(e < 1\), the conic is an ellipse.

Step 2: Use Definition of the Conic Section

The general polar equation of a conic section with a focus at the origin is \(r = \frac{ed}{1 + e\cos \theta}\), where \(d\) is the distance from the focus to the directrix. Here, the directrix \(x = -2\) means \(d = 2\).

Step 3: Substitute Values into Polar Equation

Substitute \(e = \frac{1}{4}\) and \(d = 2\) into the equation: \[ r = \frac{\frac{1}{4} \times 2}{1 + \frac{1}{4}\cos \theta} \].

Step 4: Simplify the Equation

Simplify the equation: \[ r = \frac{\frac{1}{2}}{1 + \frac{1}{4}\cos \theta} \]. Multiply the numerator and denominator by 4 to clear the fractions: \[ r = \frac{2}{4 + \cos \theta} \].

Key Concepts

Eccentricity

Eccentricity is a fundamental concept in studying conic sections, which describes how "stretched" a conic is. It is represented by the letter "e" and plays a key role in distinguishing between different types of conic sections.

- If the eccentricity is less than 1 (\(e < 1\)), the conic section is an ellipse, implying it is more circular.- When the eccentricity equals 1 (\(e = 1\)), the conic section is a parabola, exhibiting a perfect balance.- An eccentricity greater than 1 (\(e > 1\)) indicates a hyperbola, where the conic is more elongated.

In our exercise, the given eccentricity is \(e = \frac{1}{4}\). This excites us to know that we're dealing with an ellipse since \(e\) is less than 1. Eccentricity not only classifies the type of conic section but also deeply influences the shape, with smaller values leading to more circular shapes.

Ellipse

An ellipse is a type of conic section characterized by having all points equidistant to two fixed points called foci.The geometry of an ellipse is unique due to its property of having different lengths for the major and minor axes, making it appear "stretched."

• Its general polar equation, with one focus at the origin, is \( r = \frac{ed}{1+e\cos\theta} \).
• The eccentricity (\(e\)) being less than 1 ensures the curve remains closed and oval-shaped, like in our exercise where \(e = \frac{1}{4}\).
• Directrix, a line associated with the ellipse, assists in determining the shape by influencing distance and orientation.

For instance, when the directrix \(x = -2\) is used with eccentricity \(e = \frac{1}{4}\), the equation simplifies, showing us how the distances inside the ellipse relate. Thus, elliptical equations serve not only to express an ellipse's shape in polar coordinates but also to show its dependency on certain parameters like eccentricity and directrix.

Conic Sections in Polar Coordinates

Conic sections in polar coordinates are special geometric shapes whose equations vary based on eccentricity and directrix.These sections, including ellipses, parabolas, and hyperbolas, can be defined using polar equations centered around one focus, rather than the traditional Cartesian coordinate system.

• This allows a more intuitive understanding of conic properties, such as axis alignment or rotational symmetry.
• A standard polar equation for conics is \( r = \frac{ed}{1 + e \cos \theta} \), where "r" represents the radius, "e" stands for eccentricity, and \(\theta\) is the angle.
• The parameter "d" is the perpendicular distance from the directrix to the focus, influencing the resultant conic section.

In the problem given, substituting eccentricity \(e = \frac{1}{4}\) and distance \(d = 2\) into the polar equation transforms it into \( r = \frac{2}{4 + \cos \theta} \).This transformation helps convert linear measurements into a circular and sometimes rotationally symmetric coordinate system, which gives us great flexibility in analyzing and solving conic problems geometrically.