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.