Find the center, foci, vertices and asymptotes of the hyperbola (x−8)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 x2−y2=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 (8−√2,−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=±(x−8)y+6=±x∓8y=x−14 and y=−x+2
Therefore, the graph is
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