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 foci and vertices
The foci of the hyperbola are given as \((\pm 4, 0)\) and the vertices as \((\pm 2, 0)\).

### Step 2: Determine the values of \(a\) and \(c\)
From the vertices, we know:
- The distance from the center to the vertices \(a = 2\).

From the foci, we know:
- The distance from the center to the foci \(c = 4\).

### Step 3: 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 4: Write the standard form of the hyperbola
The standard form of the 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 \(b^2 = 12\):
\[ \frac{x^2}{4} - \frac{y^2}{12} = 1 \]

### Step 5: Final equation
Thus, the equation of the hyperbola is:
\[ \frac{x^2}{4} - \frac{y^2}{12} = 1 \]