Find the equation of the hyperbola whose foci are (8,3), (0,3) and eccentricity `= 4/3.`

Answer

To find the equation of the hyperbola with given foci and eccentricity, we can 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. The foci given are (8, 3) and (0, 3).

Using the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of the foci:
\[ \text{Center} = \left( \frac{8 + 0}{2}, \frac{3 + 3}{2} \right) = \left( 4, 3 \right) \]

### Step 2: Calculate the distance between the foci
The distance between the two foci is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of the foci:
\[ d = \sqrt{(8 - 0)^2 + (3 - 3)^2} = \sqrt{8^2} = 8 \]
This distance is equal to \(2c\) where \(c\) is the distance from the center to each focus. Thus:
\[ 2c = 8 \implies c = 4 \]

### Step 3: Use the eccentricity to find \(a\)
The eccentricity \(e\) of the hyperbola is given as \( \frac{4}{3} \). The relationship between \(a\), \(b\), and \(c\) for a hyperbola is:
\[ e = \frac{c}{a} \]
Substituting the known values:
\[ \frac{4}{3} = \frac{4}{a} \]
Cross-multiplying gives:
\[ 4a = 12 \implies a = 3 \]

### Step 4: Find \(b\) using the relationship between \(a\), \(b\), and \(c\)
We know:
\[ c^2 = a^2 + b^2 \]
Substituting the values of \(c\) and \(a\):
\[ 4^2 = 3^2 + b^2 \implies 16 = 9 + b^2 \implies b^2 = 16 - 9 = 7 \implies b = \sqrt{7} \]

### Step 5: Write the equation of the hyperbola
Since the foci lie on the horizontal line \(y = 3\), the hyperbola opens horizontally. The standard form of the equation of a hyperbola centered at \((h, k)\) is:
\[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \]
Substituting \(h = 4\), \(k = 3\), \(a = 3\), and \(b = \sqrt{7}\):
\[ \frac{(y - 3)^2}{3^2} - \frac{(x - 4)^2}{7} = 1 \]
This simplifies to:
\[ \frac{(y - 3)^2}{9} - \frac{(x - 4)^2}{7} = 1 \]

### Final Answer
The equation of the hyperbola is:
\[ \frac{(y - 3)^2}{9} - \frac{(x - 4)^2}{7} = 1 \]