The distance between foci of a hyperbola is 16 and its eccentricity is `sqrt2` , then the equation of hyperbola is

Options

A. `x^(2) - y^(2) = 3`
B. `x^(2) - y^(2) = 16`
C. `x^(2) - y^(2) = 15`
D. `x^(2) - y^(2) = 32`

Answer

To find the equation of the hyperbola given the distance between the foci and the eccentricity, we can follow these steps:

### Step 1: Understand the given information
We know:
- The distance between the foci (2c) = 16
- The eccentricity (e) = √2

### Step 2: Calculate the value of c
The distance between the foci is given by:
\[ 2c = 16 \]
Thus,
\[ c = \frac{16}{2} = 8 \]

### Step 3: Relate c and a using eccentricity
The relationship between the eccentricity (e), a, and c for a hyperbola is given by:
\[ e = \frac{c}{a} \]
Substituting the values we have:
\[ \sqrt{2} = \frac{8}{a} \]

### Step 4: Solve for a
Rearranging the equation gives:
\[ a = \frac{8}{\sqrt{2}} = 4\sqrt{2} \]

### Step 5: Calculate b using the relationship c² = a² + b²
We know:
\[ c^2 = a^2 + b^2 \]
Substituting the values we have:
\[ 8^2 = (4\sqrt{2})^2 + b^2 \]
\[ 64 = 32 + b^2 \]
\[ b^2 = 64 - 32 = 32 \]

### Step 6: Write the equation of the hyperbola
The standard form of the hyperbola is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Substituting the values of a² and b²:
\[ a^2 = (4\sqrt{2})^2 = 32 \]
\[ b^2 = 32 \]
Thus, the equation becomes:
\[ \frac{x^2}{32} - \frac{y^2}{32} = 1 \]

### Step 7: Simplify the equation
This can be simplified to:
\[ x^2 - y^2 = 32 \]

### Final Answer
The equation of the hyperbola is:
\[ x^2 - y^2 = 32 \]

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