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Chapter 6: Problem 31

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$x^{2}-9 y^{2}+2 x-54 y-80=0$$

Short Answer

Expert verified
The center of the hyperbola is (-1, 3). The vertices are located at (-11, 3) and (9, 3). The foci are at ((-1 - 10\sqrt{11}/3, 3) and (-1 + 10\sqrt{11}/3, 3). The equations of the asymptotes are \(y = 3 卤(10/3)(x + 1)\).

Step by step solution

01

Rewrite the equation to standard form

The standard hyperbola equation is either \((x - h)^2/a^2 - (y - k)^2/b^2 = 1\) or \((y - k)^2/a^2 - (x - h)^2/b^2 = 1\). Here, to rewrite the equation into standard form, complete the square on the x and y terms. The equation \(x^{2}-9 y^{2}+2 x-54 y-80=0\) can be rewritten as \((x + 1)^2 - 9(y - 3)^2 = 100\).
02

Find the center of the hyperbola

The center of the hyperbola is \((h, k)\). After rewriting our equation, we find that \(h = -1\) and \(k = 3\), so the center of the hyperbola is \((-1, 3)\).
03

Determine values for a, b and c

By comparing our standard equation with \(a^2 = 100\) and \(b^2 = 100/9\), we can deduce that \(a = 10\) and \(b = 10/3\). Furthermore, 'c' can be found using the relationship \(c^2 = a^2 + b^2\), which leads to \(c = \sqrt{100 + (100/9)} = \sqrt{1100/9} = 10\sqrt{11}/3\).
04

Find vertices and foci

The vertices are located at \((h - a, k)\) and \((h + a, k)\). Substituting for 'h', 'k', and 'a' we find the vertices are located at \((-11, 3)\) and \((9, 3)\). The foci are located at \((h - c, k)\) and \((h + c, k)\), so substituting our h, k, and c values, the foci are at \((-1 - 10\sqrt{11}/3, 3)\) and \((-1 + 10\sqrt{11}/3, 3)\).
05

Find the equations of the asymptotes

The equations of the asymptotes can be written as \(y - k = 卤(b/a)(x - h)\) or \(y = k 卤(b/a)(x - h)\). With our values of 'h', 'k', 'a' and 'b', the equations become \(y = 3 卤(10/3(x + 1))\).
06

Sketch the hyperbola

First, mark the center, vertices, and foci on the number plane. Next, draw the asymptotes which cross through the center with the slopes given by the equation. Finally, sketch the hyperbola approaching but never reaching the asymptotes, containing the vertices and foci within.

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