Write the equation in standard form for the hyperbola with vertices (
–
5,0) and (5,0), and a conjugate axis of length 12.

To write the equation of the hyperbola in standard form, we need to identify key elements from the given information:

1. **Vertices**: (\( -5,0 \)) and (\( 5,0 \))
2. **Conjugate axis length**: 12

The standard form of a hyperbola centered at the origin with horizontal transverse axis is:

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

### Step 1: Determine \(a\)
The distance between the vertices is \(2a\). Given that the vertices are at (\( -5,0 \)) and (\( 5,0 \)), we find \(2a = 10\), thus:

\[
a = 5
\]

### Step 2: Determine \(b\)
The length of the conjugate axis is \(2b\). Given that the conjugate axis length is 12, we find \(2b = 12\), thus:

\[
b = 6
\]

### Step 3: Write the equation
Now we can substitute \(a\) and \(b\) into the standard form equation:

\[
\frac{x^2}{5^2} - \frac{y^2}{6^2} = 1
\]

Simplifying, we get:

\[
\frac{x^2}{25} - \frac{y^2}{36} = 1
\]

Thus, the equation of the hyperbola in standard form is:

\[
\frac{x^2}{25} - \frac{y^2}{36} = 1
\]

Would you like more details or have any questions?

Here are some additional questions you might be interested in:
1. How do you derive the equation of a hyperbola given its foci?
2. What are the asymptotes of a hyperbola and how do you find them?
3. How do you convert a hyperbola's equation from general form to standard form?
4. What is the significance of the transverse and conjugate axes in a hyperbola?
5. How do you graph a hyperbola given its equation in standard form?

**Tip:** When dealing with hyperbolas, always remember to identify the center, vertices, and lengths of the transverse and conjugate axes to correctly determine the standard form equation.

Learn how to write the equation of a hyperbola in standard form using vertices and the length of the conjugate axis. This problem involves key concepts of analytic geometry and hyperbolas. Suitable for advanced high school students and beyond.

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