For $$ y^{2}-\frac{(x-7)^{2}}{18}=1 $$ (a) Identify the center, (b) Identify the vertices. (c) Identify the foci. (d) Write equations for the asymptotes. (e) Graph the hyperbola. Give exact answers. Express numbers in simplest form using integers, fractions, or radicals. Part 1 of 5 (a) The center is ( $$ \square $$ 7 . $$ \square $$ ). Part 2 of 5 (b) The vertices are $$ (7,, 1) $$ and $$ (7,-1) $$. Part 3 of 5 (c) The foci are $$ (7, \sqrt{19}) $$ and $$ (\sqrt{7},-\sqrt{19}) $$. Part: $$ 3 / 5 $$ Part 4 of 5 (d) The asymptotes are $$ y= $$ $$ \square $$ and $$ y= $$ $$ \square $$ .

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To identify the center, vertices, foci, and asymptotes of the hyperbola given by the equation $$ y^{2}-\frac{(x-7)^{2}}{18}=1 $$, we need to rewrite the equation in standard form.

The standard form of a hyperbola centered at (h, k) is given by:
$$ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 $$

Comparing the given equation with the standard form, we can see that the center of the hyperbola is (7, 0).

Now, let's identify the vertices, foci, and asymptotes:

(a) The center of the hyperbola is (7, 0).

(b) The vertices of the hyperbola are (7, 1) and (7, -1).

(c) The foci of the hyperbola are (7, √19) and (7, -√19).

(d) The asymptotes of the hyperbola are y = ±√(18/1) = ±√18 = ±3√2.

Therefore, the center is (7, 0), the vertices are (7, 1) and (7, -1), the foci are (7, √19) and (7, -√19), and the asymptotes are y = ±3√2.
Center: (7, 0) Vertices: (7, 1) and (7, -1) Foci: (7, √19) and (7, -√19) Asymptotes: y = 3√2 and y = -3√2

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