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

Find the center, foci, vertices and asymptotes of the hyperbola $\displaystyle (x - 8)^2 - (y + 6)^2 = 1$. Sketch its graph.

The shifted hyperbola has center $(8, -6)$ and a horizontal transverse axis. It is derived from the hyperbola $x^2 - y^2 = 1$ with center at the origin. Since $a^2 = 1$ and $b^2 = 1$, we have $a = 1, b = 1$ and $c = \sqrt{a^b + b^2} = \sqrt{1 + 1} = \sqrt{2}$. Thus, the foci lie $\sqrt{2}$ 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

$\displaystyle (8 + \sqrt{2},-6)$ and $(8 - \sqrt{2},-6)$

Vertices

$\displaystyle (9, -6)$ and $(7, -6)$

The asymptotes of the unshifted hyperbola are $y = \pm x$, so the asymptotes of the shifted hyperbola are


$
\begin{equation}
\begin{aligned}

y + 6 =& \pm (x - 8)
\\
\\
y + 6 =& \pm x \mp 8
\\
\\
y =& x - 14 \text{ and } y = -x + 2

\end{aligned}
\end{equation}
$



Therefore, the graph is