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

Explanation

To find the equation of the hyperbola with foci at (8, 3) and (0, 3) and an eccentricity of \( \frac{4}{3} \), 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 two foci.

\[ \text{Center} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{8 + 0}{2}, \frac{3 + 3}{2} \right) = (4, 3) \]

### Step 2: Determine the distance between the foci
The distance between the two foci is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(0 - 8)^2 + (3 - 3)^2} = \sqrt{64} = 8 \]

### Step 3: Relate the distance to the eccentricity
The distance between the foci is also given by the formula \( 2c \), where \( c = ae \) and \( e \) is the eccentricity.

Given that \( e = \frac{4}{3} \), we can express \( c \) as:

\[ c = \frac{d}{2} = \frac{8}{2} = 4 \]

### Step 4: Find the value of \( a \)
Using the relationship \( c = ae \):

\[ 4 = a \cdot \frac{4}{3} \implies a = 4 \cdot \frac{3}{4} = 3 \]

### Step 5: Find the value of \( b \)
For hyperbolas, the relationship between \( a \), \( b \), and \( c \) is given by:

\[ c^2 = a^2 + b^2 \]

Substituting the known values:

\[ 4^2 = 3^2 + b^2 \implies 16 = 9 + b^2 \implies b^2 = 16 - 9 = 7 \]

### Step 6: Write the equation of the hyperbola
Since the foci are horizontally aligned (same y-coordinate), the equation of the hyperbola is of the form:

\[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \]

Substituting \( h = 4 \), \( k = 3 \), \( a^2 = 9 \), and \( b^2 = 7 \):

\[ \frac{(x - 4)^2}{9} - \frac{(y - 3)^2}{7} = 1 \]

### Final Answer
The equation of the hyperbola is:

\[ \frac{(x - 4)^2}{9} - \frac{(y - 3)^2}{7} = 1 \]

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