Certainly! Let's find the equation of the hyperbola given its foci and vertices:
### Step 1: Identify Key Information
Given:
- Foci: [tex]\((-4,0)\)[/tex] and [tex]\((4,0)\)[/tex]
- Vertices: [tex]\((-3,0)\)[/tex] and [tex]\((3,0)\)[/tex]
From this information, we know:
- The hyperbola is centered at the origin [tex]\((0,0)\)[/tex].
- The distance [tex]\(c\)[/tex] from the center to each focus is 4.
- The distance [tex]\(a\)[/tex] from the center to each vertex is 3.
### Step 2: Write the General Form of the Hyperbola Equation
A hyperbola centered at the origin that opens horizontally has the equation:
[tex]\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\][/tex]
### Step 3: Determine [tex]\(a^2\)[/tex]
The value [tex]\(a\)[/tex] is the distance from the center to a vertex. Thus:
[tex]\[
a = 3 \implies a^2 = 3^2 = 9
\][/tex]
### Step 4: Determine [tex]\(c^2\)[/tex]
The value [tex]\(c\)[/tex] is the distance from the center to a focus. Thus:
[tex]\[
c = 4 \implies c^2 = 4^2 = 16
\][/tex]
### Step 5: Solve for [tex]\(b^2\)[/tex]
Using the relationship [tex]\(c^2 = a^2 + b^2\)[/tex], we can find [tex]\(b^2\)[/tex]:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
[tex]\[
16 = 9 + b^2
\][/tex]
Solving for [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = 16 - 9 = 7
\][/tex]
### Step 6: Write the Equation of the Hyperbola
With [tex]\(a^2 = 9\)[/tex] and [tex]\(b^2 = 7\)[/tex], the equation of the hyperbola is:
[tex]\[
\frac{x^2}{9} - \frac{y^2}{7} = 1
\][/tex]
So the final equation of the hyperbola is:
[tex]\[
\frac{x^2}{9} - \frac{y^2}{7} = 1
\][/tex]
Thus, the equation is fully determined:
[tex]\[
\frac{x^2}{9} - \frac{y^2}{7} = 1
\][/tex]
