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Chapter 11: Problem 20

Find an equation for the hyperbola described. Graph the equation. Center at (0,0)\(;\) focus at (0,5)\(;\) vertex at (0,3)

Short Answer

Expert verified
The equation is \[ \frac{y^2}{9} - \frac{x^2}{16} = 1 \]. The hyperbola opens vertically.

Step by step solution

01

Identify the hyperbola's orientation

The vertices and foci are aligned along the y-axis, indicating a vertical hyperbola.
02

Determine the values of 'a' and 'c'

The vertices are at (0, 卤3), so the distance from the center to each vertex is 'a'. Therefore, a = 3. The foci are at (0, 卤5), so the distance from the center to each focus is 'c'. Therefore, c = 5.
03

Use the relationship between 'a', 'b', and 'c'

For hyperbolas, the relationship c^2 = a^2 + b^2 holds true. Using the values identified: c^2 = a^2 + b^2 => 5^2 = 3^2 + b^2 => 25 = 9 + b^2. Solving for 'b', we get b^2 = 16, so b = 4.
04

Write the equation

For a vertical hyperbola centered at (0,0), the standard form is: \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] Substitute 'a' and 'b' values: \[ \frac{y^2}{3^2} - \frac{x^2}{4^2} = 1 \] Thus, the equation is: \[ \frac{y^2}{9} - \frac{x^2}{16} = 1 \]
05

Graph the hyperbola

Plot the vertices at (0,3) and (0,-3), and the foci at (0,5) and (0,-5). The asymptotes are lines passing through the origin with slopes 卤a/b and 卤b/a. For our equation, they are y = 卤(3/4)x. Draw the hyperbola opening upwards and downwards.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

equation of hyperbola
To derive the equation of a hyperbola, we first need to know its orientation and whether it's centered at the origin. A hyperbola oriented along the y-axis with its center at the origin has the standard equation:
\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]
Here:
  • 'a' represents the distance from the center to each vertex,
  • 'b' is derived using the relationship between 'a', 'b', and 'c',
  • 'y' and 'x' are the coordinates.
Let's break this down with an example. If the vertices are at (0, 卤3) and the foci are at (0, 卤5), we can determine 'a' and 'c':
a = 3 and c = 5, substituting these values, we have:
\[\frac{y^2}{3^2} - \frac{x^2}{b^2} = 1\]
We then solve for 'b' using the relationship c^2 = a^2 + b^2.
vertices and foci
Vertices and foci are crucial in identifying the shape and position of a hyperbola. The vertices are points where the hyperbola intersects its transverse axis, while the foci are points located along the same axis but farther away than the vertices.
In our example, the vertices are located at (0, 卤3), which means:
  • The vertex distance 'a' is 3.
The foci are located at (0, 卤5), so:
  • The focus distance 'c' is 5.
Knowing the vertices help in sketching the hyperbola's opening, and the foci help in understanding the hyperbola's stretch.
relationship between a, b, and c
In hyperbolas, the relationship between 'a', 'b', and 'c' is given by the equation: \[\text{c}^2 = \text{a}^2 + \text{b}^2\]
This equation is derived from the geometric definition of a hyperbola. Here is a breakdown for our example:
  • c = 5,
  • a = 3.
Substituting these values into the equation, we get:
\[\text{5}^2 = \text{3}^2 + \text{b}^2 \]
Solving this:
\[\text{25} = \text{9} + \text{b}^2 \]
This gives us:
  • b^2 = 16, or,
  • b = 4.
Understanding this relationship helps in finding one of the values if the other two are known.

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