Question

Find the vertices, the foci, the lengths \(a\) and \(b\) of the semiaxes, and the slope of the asymptotes for each hyperbola. Graph some. $$9 x^{2}-16 y^{2}=144$$

Short answer

Vertices: (-4,0), (4,0); Foci: (-5,0), (5,0); \(a = 4\), \(b = 3\); Slope of asymptotes: \(\pm \frac{3}{4}\).

Step-by-step solution

Identify the Hyperbola Equation

Determine if the given equation represents a horizontal or vertical hyperbola by comparing it to the standard forms \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for a horizontal hyperbola and \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) for a vertical hyperbola. In this case, the given equation \(9x^2 - 16y^2 = 144\) most closely resembles the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) after dividing everything by 144.

Rewrite in Standard Form

Divide the entire equation by 144 to rewrite it in standard form: \(\frac{x^2}{16} - \frac{y^2}{9} = 1\).

Identify and Calculate \(a^2\) and \(b^2\)

From the standard form, identify \(a^2\) and \(b^2\). In this case, \(a^2 = 16\) and \(b^2 = 9\). Then take the square root of both to find \(a\) and \(b\): \(a = 4\) and \(b = 3\).

Find the Vertices

The vertices are along the transverse axis, at a distance \(a\) from the center (0,0). For this equation, the vertices are at \((-4,0)\) and \((4,0)\).

Find the Foci

Use the relationship \(c^2 = a^2 + b^2\) to find \(c\), where \(c\) is the distance from the center to each focus. Calculating gives us \(c^2 = 16 + 9 = 25\), which means \(c = 5\). The foci are at \((-5,0)\) and \((5,0)\).

Determine the Slope of the Asymptotes

The slope of the asymptotes for a horizontal hyperbola is given by \(\pm \frac{b}{a}\). In this case, \(\pm \frac{b}{a} = \pm \frac{3}{4}\).

Sketch the Hyperbola (optional)

Plot the center, vertices, foci, and asymptotes on a graph. Draw the hyperbola approaching the asymptotes as it opens away from the center.

Key concepts

Standard Form of a Hyperbola

The standard form of a hyperbola is crucial for identifying its properties and characteristics. In mathematical terms, a hyperbola can be represented as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for a hyperbola that opens horizontally, or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) for one that opens vertically. The values \(a\) and \(b\) are the lengths of the semi-major axis and semi-minor axis respectively, which determine the shape of the hyperbola.

When transforming an equation into this form, it is essential to arrange the terms so that the \(x\)-squared and \(y\)-squared terms are on opposite sides of the equation, indicating the hyperbola’s orientation. The standard form also enables us to conveniently calculate vertices, foci, and asymptotes—the key components of a hyperbola's structure and graph.

Hyperbola Vertices Calculation

The vertices of a hyperbola are points where the curve is the widest, and they lie on the transverse axis. To calculate the vertices, you identify the value of \(a\) from the standard form of the hyperbola. The coordinates of the vertices for a hyperbola that opens horizontally are \((\pm a, 0)\), and for a hyperbola that opens vertically, they are \((0, \pm a)\).

In our example with the equation \(\frac{x^2}{16} - \frac{y^2}{9} = 1\), the value of \(a^2\) is 16. Therefore, \(a\) is 4, denoting that the vertices are 4 units away from the center along the horizontal axis, which places them at \((-4, 0)\) and \((4, 0)\).

Hyperbola Foci Calculation

The foci (singular: focus) of a hyperbola are two fixed points located along the transverse axis that are pivotal in its definition. For a hyperbola in standard form, calculating the foci involves the relationship \(c^2 = a^2 + b^2\), where \(a\) is the distance from the center to the vertices and \(c\) is the distance from the center to each focus.

Once \(a^2\) and \(b^2\) are known, we find \(c\) by taking the square root of their sum. For our example, where \(a^2 = 16\) and \(b^2 = 9\), we would compute \(c^2 = 25\), thereby getting \(c = 5\). The foci would then be located at \((-5, 0)\) and \((5, 0)\), five units from the center.

Asymptotes of a Hyperbola

The asymptotes of a hyperbola are the lines that the hyperbola approaches but never actually reaches. They represent the direction in which the two branches of the hyperbola extend to infinity and are crucial for sketching the graph accurately. For a hyperbola that opens horizontally, the equations of the asymptotes are \(y = \pm \frac{b}{a}x\), and for one that opens vertically, \(x = \pm \frac{b}{a}y\).

From our given equation, where we determined that \(a = 4\) and \(b = 3\), we calculate the slope of the asymptotes to be \(\pm \frac{3}{4}\). Therefore, the asymptotic lines will have slopes of \(3/4\) and \(\-3/4\), helping to guide the drawing of the hyperbola's curvature.