Problem 73 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 73-76, determine whether the statement is true or false. Justify your answer. In the standard form of the equation of a hyperbola, the larger the ratio of \(b\) to \(a\) the larger the eccentricity of the hyperbola.

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understand the relationship of a, b, and eccentricity in a hyperbola

The eccentricity of a hyperbola is given by the equation \(e = \sqrt{1 + (b / a)^2}\). Eccentricity is a measure of how much the hyperbola deviates from being a perfect circle.

Step 2: Analyze the given statement

The given statement suggests that as the ratio \(b / a\) increases, the eccentricity also increases. Let's analyze this by using the mathematical formula of eccentricity.

Step 3: Prove or disprove the statement

Looking at the formula for eccentricity, \(\sqrt{1 + (b / a)^2}\), it becomes obvious that as the ratio \(b / a\) increase, the eccentricity \(e\) also increases. This is because the ratio \(b / a\) is a part of the value being square-rooted, meaning an increase in this ratio will increase the total resulting value. Hence, the statement is True.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept that helps describe the shape of conic sections, including hyperbolas. A hyperbola is a type of curve on a plane, and its eccentricity ( \(e\) ) defines how much it deviates from being a perfect circle. In simpler terms, the eccentricity tells us how "stretched" the hyperbola is.

For a hyperbola, the eccentricity is always greater than 1. This is because hyperbolas are open curves, unlike circles and ellipses which have eccentricities less than or equal to 1. The formula for finding the eccentricity of a hyperbola is: \(e = \sqrt{1 + \left(\frac{b}{a}\right)^2}\) , where \(a\) and \(b\) are values derived from the hyperbola's equation.

This formula indicates that as the term \((b / a)^2\) increases, the eccentricity increases, making the hyperbola more elongated.

Standard Form of the Equation

The standard form of the equation of a hyperbola is crucial for understanding its properties and structure. Every hyperbola can be defined by an equation of the form:
\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) .

In these equations, \(a\) and \(b\) are constants that determine the shape and orientation of the hyperbola. Here, \(a\) represents the distance from the center to the vertices of the hyperbola along the axis that opens the hyperbola, while \(b\) determines the distance along the conjugate axis.

The standard form equation allows us to quickly identify these components, and understanding them is key to calculating the eccentricity and other properties of the hyperbola.

Mathematical Relationship

In mathematics, hyperbolas have equations and properties interconnected through various relationships. One vital relationship is the link between integers \(a\), \(b\), and \(e\) (eccentricity).

As stated, the eccentricity can be calculated with \(e = \sqrt{1 + \left(\frac{b}{a}\right)^2}\) , demonstrating a direct mathematical relationship between \(b/a\) and \(e\).

• As \(b / a\) increases, the value inside the square root increases.
• This, in turn, results in a higher value for the eccentricity.

Thus, any changes in the ratio \(b/a\) directly affect the eccentricity, showing how closely these elements of a hyperbola are related.

Ratio

Ratios are essential in mathematics as they represent relationships between quantities. For hyperbolas, the ratio \(b/a\) is integral in determining the eccentricity, as seen in its formula: \( e = \sqrt{1 + \left(\frac{b}{a}\right)^2} \).

This particular ratio describes the proportion between two defining components of a hyperbola: \(b\) and \(a\).

Important points about the \(b/a\) ratio:

• It affects the curve's shape and extent of stretching.
• An increase in \(b/a\) directly increases the eccentricity, making the curve more pronounced.
• It helps in understanding how the structural components of hyperbolas contribute to their overall geometry.

Understanding ratios in hyperbolas is crucial for calculating properties like the eccentricity and predicting how changes in \(a\) or \(b\) can alter the hyperbola's form.