To write the equation of the hyperbola, let's break down the problem into manageable steps.
### Step 1: Determine the Center
The center of the hyperbola is the midpoint of the line segment joining the foci. For foci [tex]\((1, -5)\)[/tex] and [tex]\((9, -5)\)[/tex]:
[tex]\[
x_{\text{center}} = \frac{1 + 9}{2} = \frac{10}{2} = 5
\][/tex]
[tex]\[
y_{\text{center}} = \frac{-5 + (-5)}{2} = \frac{-10}{2} = -5
\][/tex]
Therefore, the center of the hyperbola is [tex]\((5, -5)\)[/tex].
### Step 2: Determine the Distance Between the Foci
The distance [tex]\(2c\)[/tex] between the foci is:
[tex]\[
2c = 9 - 1 = 8 \quad \Rightarrow \quad c = \frac{8}{2} = 4
\][/tex]
### Step 3: Length of the Conjugate Axis
The length of the conjugate axis is given as 6, so:
[tex]\[
2b = 6 \quad \Rightarrow \quad b = \frac{6}{2} = 3
\][/tex]
### Step 4: Calculate [tex]\(a\)[/tex]
We know that for hyperbolas, the relationship [tex]\(c^2 = a^2 + b^2\)[/tex] holds. Here:
[tex]\[
c = 4 \quad \text{and} \quad b = 3
\][/tex]
Thus:
[tex]\[
c^2 = 4^2 = 16
\][/tex]
[tex]\[
b^2 = 3^2 = 9
\][/tex]
[tex]\[
a^2 = c^2 - b^2 = 16 - 9 = 7
\][/tex]
### Step 5: Equation of the Hyperbola
Since the foci are horizontally aligned (the y-coordinates are the same), the hyperbola opens horizontally. The standard form for a horizontally opening hyperbola is:
[tex]\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\][/tex]
Substituting the values we found:
[tex]\[
(h, k) = (5, -5), \quad a^2 = 7, \quad b^2 = 9
\][/tex]
The equation of the hyperbola is:
[tex]\[
\frac{(x - 5)^2}{7} - \frac{(y + 5)^2}{9} = 1
\][/tex]
### Conclusion
Having derived the equation from the given parameters, the closest match from the provided options is:
[tex]\[
\frac{(x-5)^2}{7} - \frac{(y+5)^2}{9} = 1
\][/tex]
This corresponds to the first option in the given question.
