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Chapter 6: Problem 44

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: \((3,0),(3,6)\) asymptotes: \(y=6-x, y=x\)

Short Answer

Expert verified
The standard form of the equation of the hyperbola is \((y-3)^2/9 - (x-3)^2/9 = 1\).

Step by step solution

01

Find the Center of the Hyperbola

To find the center of the hyperbola, find the average of the x-coordinates and the y-coordinates of the vertices separately: The x-coordinate is (3+3)/2=3. The y-coordinate is (0+6)/2=3. Thus, the center of the hyperbola is (h,k) = (3,3).
02

Find the value of a

The value of a is half the distance between the two vertices. The distance between the vertices (3,0) and (3,6) is 6 units, thus a = 6/2 = 3.
03

Find the value of b using the asymptotes

The slopes of the asymptotes for a vertical hyperbola is 卤b/a. One of the asymptote is given by y = 6 - x which is of the form y = mx + c, so m = -1. Or in other words, the slope of the asymptote = -1 = -b/a. This gives b = a = 3.
04

Formulate the Standard Equation

Substitute h, k, a, and b into the standard formula for a vertical hyperbola to get the equation: \((y - k)^2/a^2 - (x - h)^2/b^2 = 1\)) becomes \((y - 3)^2/3^2 - (x - 3)^2/3^2 = 1\). It can be further simplified to \((y-3)^2/9 - (x-3)^2/9 = 1\).

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