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Chapter 6: Problem 41

Find the standard form of the equation of the hyperbola with the given characteristics. Foci: \((0, \pm 8) ;\) asymptotes: \(y=\pm 4 x\)

Short Answer

Expert verified
The standard form of the equation of the hyperbola is \(17x^2 - 2y^2 = 544\).

Step by step solution

01

Identify the center of the hyperbola

The center of the hyperbola is at the origin (0,0) because that is the middle point between the foci and also the intersection point of the asymptotes.
02

Calculate the value of c

The distance from the center to each focus, denoted c, is 8 (the y-coordinate of the foci).
03

Calculate the slope of the asymptotes

The slope of the asymptote lines is 4, which is represented by b/a in the standard equation of a hyperbola. So, \(b=a*4\). We need to find the value of a and b.
04

Use the relationship between a, b, and c to find a and b

In a hyperbola, the relationship between a, b and c is \(c=\sqrt{a^2+b^2}\). Since we know that b=4a and c=8, we can substitute these values into the equation and solve for a. \[c=\sqrt{a^2+(4a)^2}\] \[8=\sqrt{a^2+16a^2}\] Squaring both sides, \(a=\sqrt{32/17}\), hence \(b=4a=4\sqrt{32/17}\)
05

Write down the equation of the hyperbola

The standard form of the equation of a hyperbola with center at the origin is \[x^2/a^2 - y^2/b^2 = 1\]. Substituting the values of a and b we've calculated, we get \[x^2/(32/17) - y^2/(256/17) = 1\] which simplifies to \[17x^2 - 2y^2 = 544\].

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