Wednesday, August 16, 2017

College Algebra, Chapter 8, 8.4, Section 8.4, Problem 14

Find the center, foci, vertices and asymptotes of the hyperbola (x8)2(y+6)2=1. Sketch its graph.

The shifted hyperbola has center (8,6) and a horizontal transverse axis. It is derived from the hyperbola x2y2=1 with center at the origin. Since a2=1 and b2=1, we have a=1,b=1 and c=ab+b2=1+1=2. Thus, the foci lie 2 units to the left and to the right of the center, and the vertices lie 1 unit to either side of the center. By applying transformations, we get

Foci

(8+2,6) and (82,6)

Vertices

(9,6) and (7,6)

The asymptotes of the unshifted hyperbola are y=±x, so the asymptotes of the shifted hyperbola are


y+6=±(x8)y+6=±x8y=x14 and y=x+2



Therefore, the graph is

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