To find the standard form of the equation of the hyperbola, let's follow these steps:
### Step 1: Identify the center [tex]\((h, k)\)[/tex] of the hyperbola
The center of the hyperbola is the midpoint of the line segment joining the vertices. Given the vertices [tex]\((-1, 3)\)[/tex] and [tex]\((7, 3)\)[/tex], we can find the midpoint as follows:
[tex]\[
h = \frac{-1 + 7}{2} = \frac{6}{2} = 3.0
\][/tex]
[tex]\[
k = \frac{3 + 3}{2} = \frac{6}{2} = 3.0
\][/tex]
So, the center of the hyperbola is [tex]\((h, k) = (3.0, 3.0)\)[/tex].
### Step 2: Calculate the distance between the vertices (2a)
The distance between the vertices is the difference in the x-coordinates since the y-coordinates are the same. This gives us:
[tex]\[
\text{Distance between vertices} = 7 - (-1) = 7 + 1 = 8
\][/tex]
Since [tex]\(2a\)[/tex] is the distance between the vertices:
[tex]\[
2a = 8 \implies a = \frac{8}{2} = 4.0
\][/tex]
### Step 3: Calculate the distance between the foci (2c)
Similarly, the distance between the foci is the difference in the x-coordinates:
[tex]\[
\text{Distance between foci} = 8 - (-2) = 8 + 2 = 10
\][/tex]
Since [tex]\(2c\)[/tex] is the distance between the foci:
[tex]\[
2c = 10 \implies c = \frac{10}{2} = 5.0
\][/tex]
### Step 4: Find [tex]\(b\)[/tex] using the relationship [tex]\(c^2 = a^2 + b^2\)[/tex]
We have the values of [tex]\(a\)[/tex] and [tex]\(c\)[/tex], so we can find [tex]\(b\)[/tex]:
[tex]\[
a^2 = 4.0^2 = 16.0
\][/tex]
[tex]\[
c^2 = 5.0^2 = 25.0
\][/tex]
Using the formula [tex]\(c^2 = a^2 + b^2\)[/tex]:
[tex]\[
25.0 = 16.0 + b^2 \implies b^2 = 25.0 - 16.0 = 9.0
\][/tex]
[tex]\[
b = \sqrt{9.0} = 3.0
\][/tex]
### Step 5: Write the equation of the hyperbola in standard form
The standard form of the equation of a hyperbola with a horizontal transverse axis is:
[tex]\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\][/tex]
We substitute [tex]\(h = 3.0\)[/tex], [tex]\(k = 3.0\)[/tex], [tex]\(a^2 = 16.0\)[/tex], and [tex]\(b^2 = 9.0\)[/tex]:
[tex]\[
\frac{(x - 3.0)^2}{16.0} - \frac{(y - 3.0)^2}{9.0} = 1
\][/tex]
Thus, the standard form of the equation of the hyperbola is:
[tex]\[
\left( \frac{(x - 3.0)^2}{16.0} \right) - \left( \frac{(y - 3.0)^2}{9.0} \right) = 1
\][/tex]
