Find the equation to the hyperbola whose foci, are (6,4) and (-4,4) and eccentricity is 2.

Answer

To find the equation of the hyperbola with foci at (6, 4) and (-4, 4) and an eccentricity of 2, we can follow these steps:

### Step 1: Find the center of the hyperbola
The center (h, k) of the hyperbola can be found by calculating the midpoint of the foci.

\[ h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2} \]

Where \( (x_1, y_1) = (6, 4) \) and \( (x_2, y_2) = (-4, 4) \).

\[ h = \frac{6 + (-4)}{2} = \frac{2}{2} = 1 \]
\[ k = \frac{4 + 4}{2} = \frac{8}{2} = 4 \]

Thus, the center is at (1, 4).

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

\[ 2c = \text{distance between foci} \]

Calculating the distance:

\[ 2c = \sqrt{(6 - (-4))^2 + (4 - 4)^2} = \sqrt{(6 + 4)^2 + 0^2} = \sqrt{10^2} = 10 \]

Thus, \( c = \frac{10}{2} = 5 \).

### Step 3: Find the value of 'a' using eccentricity
The eccentricity \( e \) is given by the formula:

\[ e = \frac{c}{a} \]

Given that \( e = 2 \), we can substitute to find \( a \):

\[ 2 = \frac{5}{a} \implies a = \frac{5}{2} = 2.5 \]

### Step 4: Calculate \( a^2 \)
\[ a^2 = (2.5)^2 = 6.25 \]

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

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

Substituting the known values:

\[ 5^2 = 6.25 + b^2 \implies 25 = 6.25 + b^2 \implies b^2 = 25 - 6.25 = 18.75 \]

### Step 6: Write the equation of the hyperbola
The standard form of the equation of a hyperbola centered at (h, k) is:

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

Substituting \( h = 1 \), \( k = 4 \), \( a^2 = 6.25 \), and \( b^2 = 18.75 \):

\[ \frac{(x - 1)^2}{6.25} - \frac{(y - 4)^2}{18.75} = 1 \]

### Step 7: Simplify the equation
To eliminate the fractions, we can multiply through by the least common multiple of the denominators, which is 18.75:

\[ 18.75 \cdot \frac{(x - 1)^2}{6.25} - 18.75 \cdot \frac{(y - 4)^2}{18.75} = 18.75 \]

This simplifies to:

\[ 3 \cdot (x - 1)^2 - (y - 4)^2 = 18.75 \]

### Final Equation
Thus, the equation of the hyperbola is:

\[ 3(x - 1)^2 - (y - 4)^2 = 18.75 \]