Math Problem Statement

Use the​ center, vertices, and asymptotes to graph the hyperbola. Locate the foci and find the equations of the asymptotes. left parenthesis x minus 3 right parenthesis squared minus 64 left parenthesis y plus 3 right parenthesis squaredequals64 Question content area bottom Part 1 Graph the hyperbola. Choose the correct graph below. A. -8 14 -8 2 x y

A coordinate system with a horizontal x-axis labeled from negative negative 8 to 14 in increments of 2, a vertical y-axis labeled from negative negative 8 to 2 in increments of 2. The graph of a hyperbola has one branch that opens to the right and one that opens to the left. The vertices are located at (negative 5,negative 3) and (11,negative 3). The vertical sides of a rectangle between the branches touch the branches at the vertices. The graph has center in fourth quadrant and the diagonal lines of the rectangle have magnitude of slopes between 0 and 1 and the branches approach these lines as they travel away from the center. B. -8 14 -8 2 x y

A coordinate system with a horizontal x-axis labeled from negative negative 8 to 14 in increments of 2, a vertical y-axis labeled from negative negative 8 to 2 in increments of 2. The graph of a hyperbola has one branch that opens up and one that opens down. The vertices are located at (3,negative 2) and (3,negative 4). The horizontal sides of a rectangle between the branches touch the branches at the vertices. The graph has center in fourth quadrant and the diagonal lines of the rectangle have magnitude of slopes between 0 and 1 and the branches approach these lines as they travel away from the center. C. -8 14 -8 2 x y

A coordinate system with a horizontal x-axis labeled from negative negative 8 to 14 in increments of 2, a vertical y-axis labeled from negative negative 8 to 2 in increments of 2. The graph of a hyperbola has one branch that opens to the right and one that opens to the left. The vertices are located at (11,negative 3) and (negative 5,negative 3). The vertical sides of a rectangle between the branches touch the branches at the vertices. The graph has center in fourth quadrant and the diagonal lines of the rectangle have magnitude of slopes between 0 and 1 and the branches approach these lines as they travel away from the center. D. -8 14 -8 2 x y

Solution

The given equation of the hyperbola is:

(x−3)264−(y+3)21=1\frac{(x - 3)^2}{64} - \frac{(y + 3)^2}{1} = 1

Step-by-Step Analysis:

  1. Form of the Hyperbola:

    • The hyperbola is in standard form: (x−h)2a2−(y−k)2b2=1\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 Here, h=3h = 3, k=−3k = -3, a2=64a^2 = 64, and b2=1b^2 = 1.
    • This is a horizontally opening hyperbola, as the xx-term comes first.

  2. Center:

    • The center of the hyperbola is (h,k)=(3,−3)(h, k) = (3, -3).

  3. Vertices:

    • The vertices are located a=64=8a = \sqrt{64} = 8 units from the center along the transverse axis (horizontal axis in this case).
    • Vertices: (3−8,−3)(3 - 8, -3) and (3+8,−3)(3 + 8, -3), which are: (−5,−3)(-5, -3) and (11,−3)(11, -3).

  4. Foci:

    • The foci are located c=a2+b2=64+1=65c = \sqrt{a^2 + b^2} = \sqrt{64 + 1} = \sqrt{65} units from the center along the transverse axis.
    • Foci: (3−65,−3)(3 - \sqrt{65}, -3) and (3+65,−3)(3 + \sqrt{65}, -3).

  5. Asymptotes:

    • The asymptotes of a hyperbola are determined by: y−k=±ba(x−h)y - k = \pm \frac{b}{a}(x - h) Substituting h=3h = 3, k=−3k = -3, a=8a = 8, and b=1b = 1: y+3=±18(x−3)y + 3 = \pm \frac{1}{8}(x - 3) Asymptote equations: y=−3+18(x−3)andy=−3−18(x−3).y = -3 + \frac{1}{8}(x - 3) \quad \text{and} \quad y = -3 - \frac{1}{8}(x - 3).

  6. Graph:

    • The hyperbola opens horizontally with its center at (3,−3)(3, -3), vertices at (−5,−3)(-5, -3) and (11,−3)(11, -3), and asymptotes approaching the lines y=−3+18(x−3)y = -3 + \frac{1}{8}(x - 3) and y=−3−18(x−3)y = -3 - \frac{1}{8}(x - 3).

Correct Graph:

The correct choice is A, which depicts a horizontally opening hyperbola with the described features.

Let me know if you’d like further clarification or details!
Questions to Explore:

  1. How do you derive the equations of the asymptotes for other forms of hyperbolas?
  2. How does the orientation of the hyperbola change based on the standard form of its equation?
  3. What role does a2a^2 and b2b^2 play in determining the hyperbola's features?
  4. Can you derive the distance formula for the foci of a hyperbola?
  5. How do you calculate the eccentricity of a hyperbola?

Tip: Always check whether the xx-term or yy-term comes first in a hyperbola equation—it determines its orientation.

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Math Problem Analysis

Mathematical Concepts

Conic Sections
Hyperbolas
Graphing Equations

Formulas

Standard form of a hyperbola: (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Vertices: (h ± a, k)
Foci: (h ± √(a^2 + b^2), k)
Asymptotes: y - k = ±(b/a)(x - h)

Theorems

Distance formula for foci of hyperbolas
Equation of asymptotes for hyperbolas

Suitable Grade Level

Grades 10-12

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