Problem 41 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 = √3, and b = √3.

Step 2: Find the vertices

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

Step 3: Compute the foci

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

Step 4: Compute the equations of the asymptotes

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