For a hyperbola, the foci are at `(pm 4, 0)` and vertices at `(pm 2 , 0)` . Its equation is

Options

A. `x^(2)/4 - y^(2)/12 = 1`
B. `x^(2)/12 - y^(2)/4 = 1`
C. `x^(2)/16 - y^(2)/4 = 1`
D. `x^(2)/4 - y^(2)/16 = 1`

Answer

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

### Step 1: Identify the values of \( a \) and \( c \)
The foci of the hyperbola are given at \( (\pm 4, 0) \) and the vertices at \( (\pm 2, 0) \).

From the vertices, we know:
- \( a = 2 \)

From the foci, we know:
- \( c = 4 \)

### Step 2: Use the relationship between \( a \), \( b \), and \( c \)
For hyperbolas, the relationship between \( a \), \( b \), and \( c \) is given by:
\[ c^2 = a^2 + b^2 \]

Substituting the known values:
\[ 4^2 = 2^2 + b^2 \]
\[ 16 = 4 + b^2 \]
\[ b^2 = 16 - 4 = 12 \]

### Step 3: Write the standard form of the hyperbola
The standard equation of a hyperbola centered at the origin with a horizontal transverse axis is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

Substituting \( a^2 = 4 \) and \( b^2 = 12 \):
\[ \frac{x^2}{4} - \frac{y^2}{12} = 1 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ \frac{x^2}{4} - \frac{y^2}{12} = 1 \]

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