Problem 32 - 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. $$ e=5, \quad y=-6 $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{30}{1 + 5\sin\theta} \).

Step 1: Identify the Type of Conic Section

The eccentricity \( e = 5 \) indicates that the conic is a hyperbola, since for hyperbolas, \( e > 1 \).

Step 2: Review Polar Equation of a Conic

The polar equation of a conic section with one focus at the origin is given by: \[ r = \frac{ed}{1 + e\sin\theta} \] for a vertical directrix below the focus, where \( e \) is the eccentricity and \( d \) is the distance to the directrix.

Step 3: Determine the Directrix Distance

Given that the directrix is \( y = -6 \), the distance from the origin to the directrix is \( d = 6 \). In this case, the conic opens downwards since it is a hyperbola.

Step 4: Substitute Values into the Polar Equation

Substitute \( e = 5 \) and \( d = 6 \) into the general form. We get:\[ r = \frac{5 \times 6}{1 + 5\sin\theta} \] which simplifies to: \[ r = \frac{30}{1 + 5\sin\theta} \] This is the polar equation for the given conic section.

Key Concepts

Conic Sections

Understanding conic sections starts with knowing that they are shapes created by intersecting a plane with a double-napped cone. These include circles, ellipses, parabolas, and hyperbolas.
Each of these shapes has unique characteristics. For instance, a circle is a set of all points equidistant from a center point. On the other hand, an ellipse looks like a stretched circle. Parabolas form a U-like shape, commonly seen in satellite dishes and suspension bridges. And finally, hyperbolas appear as two separate curves, often resembling an "X" shape.
Conic sections are an important part of geometry and appear in various real-world applications, from architecture to astronomy. Learning about them paves the way for understanding more complex mathematical concepts.

Eccentricity

Eccentricity is a critical concept when studying conic sections. It measures how much a conic section deviates from being circular. The eccentricity, denoted as \(e\), helps distinguish between the different types of conics.

• For circles, the eccentricity \(e = 0\).
• An ellipse has an eccentricity between 0 and 1 (\(0 < e < 1\)).
• Parabolas have an eccentricity exactly equal to 1 (\(e = 1\)).
• Hyperbolas are characterized by an eccentricity greater than 1 (\(e > 1\)).

This value of \(e\) determines the nature of the conic section. In the given exercise, the eccentricity \(e = 5\) clearly indicates that the shape is a hyperbola. By understanding eccentricity, you can predict the type of conic and its general form, making it a vital tool in geometry.

Hyperbolas

Hyperbolas are fascinating conic sections that appear when a plane cuts both nappes of the double cone. Unlike ellipses and parabolas, hyperbolas consist of two disconnected curves. These curves are symmetrical and resemble an elongated "X".
Each hyperbola has two foci, just like an ellipse, and these foci help define its shape. The concept of eccentricity is crucial here because, for hyperbolas, the eccentricity \(e\) is always greater than 1. This property distinguishes hyperbolas from other conics.
The polar equation of a hyperbola is given in the form \( r = \frac{ed}{1 + e\sin\theta} \) (or \( r = \frac{ed}{1 + e\cos\theta} \), which depends on the orientation). Understanding this helps us convert characteristics like eccentricity and directrix into a polar equation, as seen in the example problem.
Hyperbolas are encountered in various areas, such as physics, engineering, and navigation systems, thus making them significant in practical scenarios.