Hyperbolas

Exercises 27-34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch.

8x² − 2y² = 16

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Start by rewriting the given equation \(8x^{2} - 2y^{2} = 16\) to isolate the terms and express it in the standard form of a hyperbola. Divide both sides of the equation by 16 to normalize the right side to 1, which is typical for conic sections.

After dividing, the equation becomes \(\frac{8x^{2}}{16} - \frac{2y^{2}}{16} = 1\). Simplify the fractions to get \(\frac{x^{2}}{2} - \frac{y^{2}}{8} = 1\). This is the standard form of a hyperbola centered at the origin with the transverse axis along the x-axis.

Identify the values of \(a^{2}\) and \(b^{2}\) from the standard form. Here, \(a^{2} = 2\) and \(b^{2} = 8\). These values will help in finding the asymptotes and foci.

Write the equations of the asymptotes for the hyperbola. Since the transverse axis is horizontal, the asymptotes are given by \(y = \pm \frac{b}{a} x\). Substitute \(a = \sqrt{2}\) and \(b = \sqrt{8}\) to express the asymptotes explicitly.

To find the foci, calculate \(c\) using the relationship \(c^{2} = a^{2} + b^{2}\). Then, the foci are located at \((\pm c, 0)\) because the transverse axis is along the x-axis. Use these points along with the asymptotes to sketch the hyperbola.

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Standard Form of a Hyperbola

The standard form of a hyperbola equation is either (x-h)²/a² - (y-k)²/b² = 1 or (y-k)²/a² - (x-h)²/b² = 1, where (h, k) is the center. Converting the given equation into this form helps identify key features like the center, vertices, and orientation of the hyperbola.

Asymptotes of a Hyperbola

Asymptotes are straight lines that the hyperbola approaches but never touches. For a hyperbola centered at (h, k), the asymptotes have equations y = k ± (b/a)(x - h) or y = k ± (a/b)(x - h), depending on the hyperbola's orientation. Finding asymptotes aids in sketching the hyperbola accurately.

Foci of a Hyperbola

The foci are two fixed points located along the transverse axis, used to define the hyperbola. Their distance from the center is given by c, where c² = a² + b². Identifying the foci helps understand the hyperbola's shape and is essential for accurate graphing.

HyperbolasExercises 27-34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch.8x² − 2y² = 16