Problem 47 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The eccentricity is \(\frac{\sqrt{10}}{3}\).

Step 1: Identify the Type of Conic Section

The given equation \(x^2 - 9y^2 = 18\) can be identified as a hyperbola since the terms \(x^2\) and \(y^2\) have opposite signs.

Step 2: Write the Standard Form of the Hyperbola Equation

The standard form of a hyperbola equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). To achieve this, divide the given equation \(x^2 - 9y^2 = 18\) by 18. We get \(\frac{x^2}{18} - \frac{y^2}{2} = 1\). Here, \(a^2 = 18\) and \(b^2 = 2\).

Step 3: Determine the Eccentricity Formula for a Hyperbola

For a hyperbola, the eccentricity \(e\) is given by the formula \(e = \sqrt{1 + \frac{b^2}{a^2}}\).

Step 4: Plug Values into the Eccentricity Formula

Since \(a^2 = 18\) and \(b^2 = 2\), substitute into the formula: \(e = \sqrt{1 + \frac{2}{18}}\).

Step 5: Simplify the Expression to Find Eccentricity

Calculate \(\frac{2}{18} = \frac{1}{9}\). Thus, \(e = \sqrt{1 + \frac{1}{9}} = \sqrt{\frac{10}{9}}\). Simplify to find \(e = \frac{\sqrt{10}}{3}\).

Key Concepts

Hyperbola

A hyperbola is a fascinating shape formed by the intersection of a plane with a double cone. It is part of the family of conic sections, which also includes ellipses, parabolas, and circles. Here's why it's interesting:

• A hyperbola consists of two separate curves that are mirror images of each other.
• The curves open away from each other, unlike an ellipse where they form a single loop.
• Hyperbolas have two axes of symmetry: the transverse and the conjugate axis.

When dealing with hyperbolas, their unique property is that the difference in distances from any point on the curve to two fixed points (foci) is constant. This is key to identifying and understanding their behavior in mathematical problems.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. The most common conic sections include:

• Circle: A special type of ellipse where both axes are equal.
• Ellipse: An oval shape formed when the plane intersects the cone at an angle less than the opening angle of the cone.
• Parabola: A curve formed when the plane is parallel to the edge of the cone.
• Hyperbola: Formed when the plane cuts through both nappes of the cone.

Understanding the properties and derivations of each shape can greatly aid in applying them to various problems, such as those involving the reflection of light or satellite dish design. Each of these conic sections has its own equations and characteristics, making them versatile tools in mathematics.

Standard Form

The standard form of an equation is a way of writing it to reveal its key properties and simplify calculations. For hyperbolas, this is especially helpful because it lays out all necessary components clearly. The general standard form of a hyperbola equation is:\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] This form allows mathematicians to easily identify the coefficients, determine the axis orientation, and calculate further properties like the foci and asymptotes. In our exercise, converting the given equation to its standard form was essential to find eccentricity, as it clarified which values to use for further calculations.

Mathematical Formula

Mathematical formulas are the backbone of solving equations, and understanding how to use them is crucial. In our exercise, we dealt with the formula for determining the eccentricity of a hyperbola:\[ e = \sqrt{1 + \frac{b^2}{a^2}} \] This formula helps in quantifying how "stretched" the conic section is. For hyperbolas, the eccentricity is always greater than 1. Smaller values close to 1 indicate a shape closer to a rectangular hyperbola, whereas larger values signify a more elongated form. Understanding which formula to apply, and how to derive values from it, is key to learning successful problem-solving techniques in math.