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Chapter 6: Problem 20

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sketch the hyperbola using the asymptotes as an aid. $$\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$$

Short Answer

Expert verified
The center of the hyperbola is at (0,0), vertices at (卤3, 0), foci at (卤鈭�34, 0) and the equations of the asymptotes are \(y = 卤 \frac{5}{3}x\).

Step by step solution

01

Identifying the Center

The standard form of a hyperbola is \(\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1\) or \(\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1\), where \(h, k\) is the center. In the equation \(\frac{x^{2}}{9}-\frac{y^{2}}{25}=1\), h and k are 0. Hence, the center of the hyperbola is at (0,0).
02

Identifying Vertices and Orientation

The distance from the center to a vertex along the hyperbola's axis is 'a'. The value of 'a' is the square root of the denominator under \(x^{2}\) as the x term is positive and thus, the transverse axis will run parallel to the x-axis. Hence, 'a' is 3 and the vertices will be at (卤3, 0).
03

Identifying Foci

The distance from the center to a focus is given by \(c = \sqrt{a^{2} + b^{2}}\), where 'b' is the square root of the denominator under \(y^{2}\). In the given equation, 'b' is 5. Substituting 'a' and 'b' in the formula, c equals \(\sqrt{3^{2} + 5^{2}} = \sqrt{9 + 25} = \sqrt{34}\). So the foci are at (卤鈭�34, 0).
04

Identifying the Asymptotes

The equations of the asymptotes of the hyperbola are given by \(y = 卤 \frac{b}{a}x\). Hence, the equations of the asymptotes are \(y = 卤 \frac{5}{3}x\).
05

Sketching the Hyperbola

Plot the center, vertices, and foci on a graph. Sketch the two asymptote lines as diagonals of a rectangle with sides of lengths 2a and 2b for guidance. The two branches of the hyperbola approach these lines as \(x 鈫� 卤 鈭瀄). Hence, completing the graph of the hyperbola.

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Most popular questions from this chapter

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{6}{2-3 \sin \theta}$$

In Exercises \(129-132,\) determine whether the statement is true or false. Justify your answer. If \(\theta_{1}=\theta_{2}+2 \pi n\) for some integer \(n,\) then \(\left(r, \theta_{1}\right)\) and \(\left(r, \theta_{2}\right)\) represent the same point in the polar coordinate system.

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$r=-3 \sin \theta$$

Determine whether the statement is true or false. Justify your answer. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical.

Consider a line with slope \(m\) and \(y\) -intercept \((0,4)\) (a) Write the distance \(d\) between the origin and the line as a function of \(m\) (b) Graph the function in part (a). (c) Find the slope that yields the maximum distance between the origin and the line. (d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.
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