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Chapter 6: Problem 73

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.

Short Answer

Expert verified
The statement is true.

Step by step solution

01

Understand the concept of eccentricity

The eccentricity (\(e\)) of a hyperbola is given by the equation \(e = \sqrt{1 + \frac{b^2}{a^2}}\). Here, \(a\) and \(b\) are constants that define the shape of the hyperbola. Eccentricity describes the shape of the conic section, for a hyperbola, it is always greater than 1.
02

Understand the impact of the ratio \(b/a\) on eccentricity

Looking at the eccentricity formula, if the ratio of \(b\) to \(a\) (i.e. \(b/a\)) increases while \(a\) remains constant, the value under the square root will increase, hence the eccentricity will increase as it's directly proportional to the value under the square root.
03

Verification of Statement

Given the direct dependency of the eccentricity on the values of \(a\) and \(b\), the statement 'The larger the ratio of \(b\) to \(a\), the larger the eccentricity of the hyperbola' is true.

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