Math Problem Statement

Find the coordinates and graph the hyperbola defined by the equation \(\frac{x^2}{100} - \frac{y^2}{81} = 1\).

Solution

The equation given in the image is:

x2100y281=1\frac{x^2}{100} - \frac{y^2}{81} = 1

This is the standard form of a hyperbola. The general form of a hyperbola centered at the origin is:

x2a2y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

From comparing the two equations, we have:

  • a2=100a^2 = 100, so a=10a = 10
  • b2=81b^2 = 81, so b=9b = 9

Important Features of the Hyperbola:

  • Center: (0,0)(0, 0) because the equation is in its standard form and there is no translation.
  • Vertices: The vertices are located along the x-axis at (±a,0)(\pm a, 0), which in this case gives the vertices at (±10,0)(\pm 10, 0).
  • Co-vertices: The co-vertices are along the y-axis at (0,±b)(0, \pm b), which gives (0,±9)(0, \pm 9). These help in defining the rectangle that guides the asymptotes.
  • Asymptotes: The asymptotes of the hyperbola are lines that pass through the center, with slopes given by ±ba\pm \frac{b}{a}. The asymptotes are: y=±910xy = \pm \frac{9}{10}x

Graphing the Hyperbola:

  1. Start by plotting the center at (0,0)(0, 0).
  2. Plot the vertices at (10,0)(10, 0) and (10,0)(-10, 0).
  3. Plot the co-vertices at (0,9)(0, 9) and (0,9)(0, -9).
  4. Draw a rectangle using the vertices and co-vertices as guides. The diagonals of this rectangle are the asymptotes of the hyperbola.
  5. Sketch the hyperbola, which opens horizontally since the positive term is with x2x^2, and approaches the asymptotes as it moves away from the center.

Would you like any further details or explanations?

Follow-up questions:

  1. How do we determine if a hyperbola opens horizontally or vertically?
  2. Can we apply the same method if the hyperbola is not centered at the origin?
  3. How do asymptotes help in sketching a hyperbola?
  4. What is the significance of the foci of a hyperbola?
  5. How would the equation change if the hyperbola opened vertically?

Tip: When graphing conic sections like hyperbolas, identifying the center, vertices, and asymptotes is crucial to sketching an accurate graph.

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Math Problem Analysis

Mathematical Concepts

Conic Sections
Hyperbolas
Coordinate Geometry

Formulas

Standard form of a hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
Vertices: \((\pm a, 0)\) or \((0, \pm b)\)
Asymptotes: \(y = \pm \frac{b}{a}x\)

Theorems

Properties of Hyperbolas

Suitable Grade Level

Grades 10-12