Chapter 10: Q. 19 (page 667)

In Problems 19–28, find an equation for the hyperbola described. Graph the equation by hand. Center at $(0, 0)$ ; focus at $(3, 0)$ ; vertex at $(1, 0)$ .

Short Answer

Expert verified

Equation of the hyperbola is: $x^{2} - \frac{y^{2}}{8} = 1$

Graph of the hyperbola is:

Step by step solution

Step 1. Given information

Hyperbola with Center at $(0, 0)$ ; focus at $(3, 0)$ ; vertex at $(1, 0)$ .

Step 2. Required to find

Find the equation of the hyperbola. Graph the equation.

Step 3. Finding the equation of the hyperbola

Given, Hyperbola with Center at $(0, 0)$ ; focus at $(3, 0)$ ; vertex at $(1, 0)$ .

Comparing with standard form of hyperbola, $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1, b^{2} = c^{2} - a^{2}; foci = (\pm c, 0); vertices = (\pm a, 0)$

We get, $a = 1, c = 3$

$b^{2} = c^{2} - a^{2} \Rightarrow b^{2} = 3^{2} - 1^{2} \Rightarrow b^{2} = 9 - 1 \Rightarrow b^{2} = 8 \Rightarrow b = \pm \sqrt{8}$

Substituting $a$ and $b$ in the equation,

$\frac{x^{2}}{1} - \frac{y^{2}}{8} = 1 \Rightarrow x^{2} - \frac{y^{2}}{8} = 1$

Step 4. Graphing the hyperbola

To determine the opening of hyperbola, substitute $x = \pm 3$ in the equation and find $y$ .

${(\pm 3)}^{2} - \frac{y^{2}}{8} = 1 \Rightarrow \frac{y^{2}}{8} = 9 - 1 \Rightarrow y^{2} = 64 \Rightarrow y = \pm 8$

The graph of the hyperbola is:

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In Problems 19–28, find an equation for the hyperbola described. Graph the equation by hand. Center at $(0, 0)$ ; focus at $(3, 0)$ ; vertex at $(1, 0)$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Required to find

Step 3. Finding the equation of the hyperbola

Step 4. Graphing the hyperbola

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In Problems 19–28, find an equation for the hyperbola described. Graph the equation by hand. Center at (0,0); focus at (3,0); vertex at(1,0).

Short Answer

Step by step solution

Step 1. Given information

Step 2. Required to find

Step 3. Finding the equation of the hyperbola

Step 4. Graphing the hyperbola

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Pure Maths

Decision Maths

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 19–28, find an equation for the hyperbola described. Graph the equation by hand. Center at $(0, 0)$ ; focus at $(3, 0)$ ; vertex at $(1, 0)$ .