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Chapter 7: Problem 41

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$(x-1)^{2}-(y-2)^{2}=3$$

Short Answer

Expert verified
The center for the hyperbola is at point (1, 2). The vertices are located at points (1-鈭�3, 2) and (1+鈭�3, 2). The foci are at (1-鈭�6, 2) and (1+鈭�6, 2). The equations of the asymptotes are y = 2 卤 (x - 1).

Step by step solution

01

Identify the center, a, and b from the given equation

Rewrite the given equation into standard form of hyperbola (x-h)^{2}/a^{2} - (y-k)^{2}/b^{2} = 1. From here, (h, k) gives the coordinates of the center. a^2 is the term associated with x in the denominator, and b^2 is the term associated with y. In this case, the equation (x-1)^{2} - (y-2)^{2} = 3 can be rewritten as \((x-1)^{2}/3 - (y-2)^{2}/3 = 1\). Therefore, the center for the hyperbola is (1, 2), a = 鈭�3, and b = 鈭�3.
02

Find the vertices

The vertices for hyperbola are given by (h卤a, k). Substituting h = 1, k = 2, and a = 鈭�3, we obtain the vertices to be at (1-鈭�3, 2) and (1+鈭�3, 2)
03

Compute the foci

The distance from the center to a focus is given by c, where \(c = 鈭�(a^{2} + b^{2})\). Substituting values, \(c = 鈭�((3) + (3)) = 鈭�6\). The foci are (h卤c, k), thus the coordinates of the foci are (1-鈭�6, 2) and (1+鈭�6, 2).
04

Compute the equations of the asymptotes

The equations of the asymptotes are given by: y = k 卤 (b/a)(x - h). Here, a = b = 鈭�3, h = 1, k = 2, hence the equations of the asymptotes are: y = 2 卤 ((鈭�3 / 鈭�3)(x - 1)) = 2 卤 (x - 1).

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