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Chapter 6: Problem 20

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. $$9 x^{2}-y^{2}+54 x+10 y+55=0$$

Short Answer

Expert verified
The center of the hyperbola is at (-3, 5), the vertices are at (-4,5) and (-2,5), the foci are at (-3卤sqrt(26), 5) and the equations of the asymptotes are y= 5卤5(x+3).

Step by step solution

01

Re-writing the equation in standard form:

To write the given equation in standard form, complete the square on x and y. The standard form for a hyperbola that opens left-right is \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\). To have this, rearrange and group x and y terms: \(9(x^2+6x)+(-y^2+10y)=-55\). Then, rewrite it as squared binomials: \(9[(x+3)^2-9] - [(y-5)^2-25]=-55\), which simplifies further to: \((x+3)^2/1^2 - (y-5)^2/5^2 = 1\).
02

Identifying properties of the hyperbola:

Here, according to the standard form: h=-3, k=5, so the center is (-3, 5). The distance from center to vertices is given by a=1 and the distance from center to co-vertices is given by b=5. Calculate c using the equation c=sqrt(a^2 + b^2), which yields c=sqrt(26). Therefore, the vertices are (-3卤1, 5) i.e. (-4,5) and (-2,5); the foci are (-3卤sqrt(26), 5).
03

Writing the equations of asymptotes:

The equations for the asymptotes of a hyperbola in standard form is given by \(y=k卤(b/a)(x-h)\). Thus, substituting values, the equations are: y= 5卤5(x+3)
04

Graphing the hyperbola and its asymptotes:

With the center, the vertices, the foci and the equations of the asymptotes, a graphing utility can be used to appropriately graph the hyperbola. Note that the foci and centers should align along the transverse axis of the hyperbola.

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Most popular questions from this chapter

In Exercises 25-28, use a graphing utility to graph the polar equation. Identify the graph. $$ r=\frac{5}{-4+2 \cos \theta} $$

Sketch the graph of each equation. (a) \(r=3 \sec \theta\) (b) \(r=3 \sec \left(\theta-\frac{\pi}{4}\right)\) (c) \(r=3 \sec \left(\theta+\frac{\pi}{3}\right)\) (d) \(r=3 \sec \left(\theta-\frac{\pi}{2}\right)\)

In Exercises 29-32, use a graphing utility to graph the rotated conic. $$ r=\frac{6}{2+\sin (\theta+\pi / 6)} $$

\(r^{2}=16 \cos 2 \theta\)

\(r=16 \cos 3 \theta\)
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