Problem 67
Question
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
Verified Answer
The statement is true; the larger the ratio of \(b\) to \(a\), the larger the eccentricity in a hyperbola.
1Step 1: Understand Eccentricity
In a hyberbola, the eccentricity, denoted as \(e\), represents how much the hyperbola deviates from being a perfect circle. It is provided by the formula \(e = \sqrt{1+\frac{b^2}{a^2}}\), where \(a\) and \(b\) represent lengths related to the size of the hyperbola.
2Step 2: Substitute and Analyze Ratio
To analyze the claim that a larger \(b/a\) ratio increases the eccentricity, let's substitute \(b=ka\) into the eccentricity formula, where \(k\) is the \(b/a\) ratio. This gives us \(e = \sqrt{1+k^2}\). From this formula, it's clear that as \(k\) (which is \(b/a\)) increases, the value under the square root increases, subsequently the eccentricity \(e\) increases.
3Step 3: Conclusion
Since the eccentricity \(e\) increases as the ratio \(b/a\) increases, we can conclude that the statement is true
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