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Chapter 6: Problem 42

Find the standard form of the equation of the hyperbola with the given characteristics. $$\text { Foci: }(\pm 10,0) ; \text { asymptotes: } y=\pm \frac{3}{4} x$$

Short Answer

Expert verified
The standard form of the equation of the hyperbola is \(x^2/100 - y^2/56.25=1\).

Step by step solution

01

Find a

The distance between the foci of a hyperbola is \(2a\), and we know the foci are at \(\pm 10,0\), so \(2a=20\). Solving for a gives us \(a=10\).
02

Find b

The slope of the asymptotes for a hyperbola is \(\pm b/a\). We were given the equation of the asymptotes as \(y = \pm \frac{3}{4}x\), and we already know from step 1 that \(a=10\). Thus, we can set \(b/a = 3/4\), and solve for b. We get \(b = \frac{3}{4} \times a\), so \(b = \frac{3}{4} \times 10\), which gives \(b=7.5\).
03

Formulate the equation of the hyperbola.

Now that we have the values for a and b, we can write down the equation of the hyperbola. Since the foci are along the x-axis (at \(\pm 10,0\)), we know this is a horizontal hyperbola. The standard form of a horizontal hyperbola is \(x^2/a^2 - y^2/b^2 = 1\), so substituting our values for a and b we get \(x^2/10^2 - y^2/7.5^2 = 1\), or simplified, \(x^2/100 - y^2/56.25 =1\).

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