To find the equation of the hyperbola with foci at the points \((-3, 5)\) and \((5, 5)\) and a length of latus rectum of \(2\sqrt{8}\) units, we will follow these steps:
### Step 1: Find the center of the hyperbola
The center of the hyperbola is the midpoint of the line segment joining the foci.
\[
\text{Center} (h, k) = \left( \frac{-3 + 5}{2}, \frac{5 + 5}{2} \right) = \left( \frac{2}{2}, 5 \right) = (1, 5)
\]
### Step 2: Calculate the distance between the foci
The distance between the foci is given by the formula \(2c\), where \(c\) is the distance from the center to each focus.
Using the distance formula:
\[
2c = \sqrt{(-3 - 5)^2 + (5 - 5)^2} = \sqrt{(-8)^2 + 0} = \sqrt{64} = 8
\]
Thus, \(c = \frac{8}{2} = 4\).
### Step 3: Find the value of \(a\) using the length of the latus rectum
The length of the latus rectum \(L\) is given by the formula:
\[
L = \frac{2b^2}{a}
\]
Given that \(L = 2\sqrt{8}\), we can set up the equation:
\[
2b^2 = 2\sqrt{8} \cdot a \implies b^2 = \sqrt{8} \cdot a
\]
### Step 4: Relate \(a\), \(b\), and \(c\)
For hyperbolas, we have the relationship:
\[
c^2 = a^2 + b^2
\]
Substituting \(c = 4\):
\[
16 = a^2 + b^2
\]
### Step 5: Substitute \(b^2\) in terms of \(a\)
From the previous step, we substitute \(b^2\):
\[
b^2 = \sqrt{8} \cdot a
\]
Substituting this into the equation \(16 = a^2 + b^2\):
\[
16 = a^2 + \sqrt{8} \cdot a
\]
Rearranging gives us:
\[
a^2 + \sqrt{8}a - 16 = 0
\]
### Step 6: Solve the quadratic equation for \(a\)
Using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(A = 1\), \(B = \sqrt{8}\), and \(C = -16\).
\[
a = \frac{-\sqrt{8} \pm \sqrt{(\sqrt{8})^2 - 4 \cdot 1 \cdot (-16)}}{2 \cdot 1}
\]
Calculating the discriminant:
\[
(\sqrt{8})^2 = 8 \quad \text{and} \quad -4 \cdot 1 \cdot (-16) = 64 \implies 8 + 64 = 72
\]
Thus:
\[
a = \frac{-\sqrt{8} \pm \sqrt{72}}{2}
\]
Calculating \(\sqrt{72} = 6\sqrt{2}\):
\[
a = \frac{-\sqrt{8} \pm 6\sqrt{2}}{2}
\]
Choosing the positive root (since \(a\) must be positive):
\[
a = \frac{-\sqrt{8} + 6\sqrt{2}}{2}
\]
### Step 7: Calculate \(b^2\)
Substituting \(a\) back into \(b^2 = \sqrt{8} \cdot a\).
### Step 8: Write the equation of the hyperbola
The standard form of the hyperbola is:
\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\]
Substituting \(h = 1\), \(k = 5\), \(a^2\), and \(b^2\) into the equation.
### Final Equation
After calculating \(a^2\) and \(b^2\), we can write the final equation of the hyperbola.
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