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Chapter 10: Problem 44

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

Short Answer

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

Step by step solution

01

Find semi-transverse axis (a)

For a hyperbola, the distance between the foci is given by \(2ae\) where \(a\) is the semi-transverse axis and \(e\) is the eccentricity. Since the foci are at \((卤10,0)\), the distance between the foci is \(20\). But since \(e\) for a hyperbola is always more than 1, \(2ae > 2a\), and so, \(a 鈮� 10\). Thus, \(a = 10\).
02

Determine semi-conjugate axis (b)

The slope of the asymptotes of a hyperbola is 卤b/a. Given equations of the asymptotes are \(y = 卤 \frac{3}{4}x\), so b/a is equal to \( 3/4\). We鈥檝e found a to be \(10\) in the first step, so: \( b/a = 3/4 \rightarrow b = 10* 3/4 = 7.5 \)
03

Write the Standard Form of the Equation

Insert the calculated values of a and b into the standard equation of the hyperbola. Since the hyperbola is opening to the sides due to the location of the foci, the standard form of the equation of the hyperbola is \(x^2/a^2 - y^2/b^2 = 1\). Replacing \( a = 10 \) and \( b = 7.5 \), this becomes \(x^2/10^2 - y^2/7.5^2 = 1\).

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