Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Find the eccentricity e of each ellipse or hyperbola. $$x^{2}-9 y^{2}=18$$

Step-by-Step Solution

Answer

The eccentricity \(e\) of the hyperbola is \(\frac{\sqrt{10}}{3}\).

Step 1: Equation Rewrite

The given equation is not in the standard form of a hyperbola. Start by rewriting it as \(\frac{x^2}{18} - \frac{9y^2}{18} = 1\), which simplifies to \(\frac{x^2}{18} - \frac{y^2}{2} = 1\).

Step 2: Identify Standard Hyperbola Form

The standard form of a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Comparing with our equation \(\frac{x^2}{18} - \frac{y^2}{2} = 1\), identify \(a^2 = 18\) and \(b^2 = 2\). Thus, \(a = \sqrt{18} = 3\sqrt{2}\) and \(b = \sqrt{2}\).

Step 3: Calculate Eccentricity

For a hyperbola, the eccentricity \(e\) is calculated as \(e = \sqrt{1 + \frac{b^2}{a^2}}\). Substitute the values for \(a^2\) and \(b^2\): \(e = \sqrt{1 + \frac{2}{18}} = \sqrt{1 + \frac{1}{9}} = \sqrt{\frac{10}{9}} = \frac{\sqrt{10}}{3}\).

Key Concepts

Hyperbola

A hyperbola is a type of conic section that forms an open curve with two distinct branches. It is defined by its geometric shape, which is created when a double cone is intersected by a plane in such a way that the angle of intersection is more acute than the angle between the cone's side and axis. Hyperbolas have two focal points, and each point on the hyperbola has a constant difference in distance to these focal points. This unique property of hyperbolas is pivotal to understanding their behavior and mathematical representation. In practical terms, hyperbolas can often describe natural phenomena, such as the path of a satellite as it escapes the gravitational pull of a planet. They are important in fields like physics and engineering, where relationships between variables might not be linear, but rather follow more complex paths.

Standard Form of a Hyperbola

Understanding the standard form of a hyperbola is crucial for analyzing its characteristics. A hyperbola typically has two branches, and its standard form equation helps us understand its orientation and properties.The standard form of a hyperbola's equation is:\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] or, in its other form, based on orientation:\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]Here,

• \(a\) represents the distance from the center to a vertex along the x-axis or y-axis, depending on orientation.
• \(b\) signifies the distance along the other axis.

The values \(a^2\) and \(b^2\) help in identifying the stretches of the hyperbola along its principal axes. The equation can be oriented in different ways, either horizontally (opening left and right) or vertically (opening up and down), affecting how it is solved.

Eccentricity Calculation

Eccentricity is a measure of how "stretched" a hyperbola is. For hyperbolas, the eccentricity is always greater than 1. This parameter helps in understanding how the two branches of the hyperbola are separated in space.The eccentricity \(e\) of a hyperbola is calculated using the formula:\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]Let’s dissect the given values in our exercise:

• Start with \(a^2 = 18\) and \(b^2 = 2\).
• Plug these into the formula: \(e = \sqrt{1 + \frac{2}{18}} = \sqrt{1 + \frac{1}{9}} = \sqrt{\frac{10}{9}} = \frac{\sqrt{10}}{3}\).

Notice how the eccentricity calculation reshapes our understanding of the hyperbola's structure, showing its characteristic wide opening compared to other conic sections like ellipses.

Conic Sections

Conic sections are the shapes created by slicing a double cone (a cone that extends upward and downward) in different ways. The main types of conic sections are circles, ellipses, parabolas, and hyperbolas. Understanding conic sections is crucial because they describe a wide range of phenomena in geometry and real-world applications. Each type has unique properties:

• A circle is formed by a plane intersecting the cone parallel to its base, creating a perfect round shape.
• An ellipse occurs when a plane intersects the cone at an angle, yet does not pass through the base, creating an oval shape.
• A parabola results from a plane that is parallel to the slant of the cone, causing a mirror-symmetrical open curve.
• A hyperbola is formed when a plane intersects both halves of the cone, resulting in two opposite-opening curve branches.

These shapes are foundational in the study of geometry, precalculus, and calculus. Each provides a lens through which to view various physical and theoretical problems, with hyperbolas often being employed in fields requiring modeling of open trajectories or waves. Conic sections are not only mathematical constructs but also practical tools used to understand the world.