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Chapter 10: Problem 67

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; the larger the ratio of \(b\) to \(a\), the larger the eccentricity in a hyperbola.

Step by step solution

01

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.
02

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.
03

Conclusion

Since the eccentricity \(e\) increases as the ratio \(b/a\) increases, we can conclude that the statement is true

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Most popular questions from this chapter

In your own words, define the term eccentricity and explain how it can be used to classify conics. Then explain how you can use the values of \(b\) and \(c\) to determine whether a polar equation of the form $$r=\frac{a}{b+c \sin \theta}$$ represents an ellipse, a parabola, or a hyperbola.

Determine whether the statement is true or false. Justify your answer. If \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) represent the same point in the polar coordinate system, then \(\left|r_{1}\right|=\left|r_{2}\right|\)

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{-3}{-4+2 \cos \theta}$$

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{5}{1-\sin \theta}$$

Determine whether the statement is true or false. Justify your answer. The point which lies on the graph of a parabola closest to its focus is the vertex of the parabola.
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