One focus at (4,0) corresponding directrix x = 1 .

Options

A. `4x^(2) - 12y^(2) = 1`
B. `125x^(2) - 100y^(2) = 9`
C. `9x^(2) -9y^(2) = 100`
D. `3x^(2) - y^(2) = 12`

Answer

To solve the problem of finding the equation of the hyperbola given one focus at (4, 0) and the corresponding directrix x = 1, we will follow these steps:

### Step 1: Identify the focus and directrix
The focus is given as \( F(4, 0) \) and the directrix is given as \( x = 1 \).

### Step 2: Use the definition of a hyperbola
The definition of a hyperbola states that the difference in distances from any point on the hyperbola to the two foci is constant. However, in this case, we can also use the relationship between the focus, directrix, and eccentricity \( e \).

### Step 3: Set up equations for the focus and directrix
Let \( A \) be the distance from the center to the focus, and \( e \) be the eccentricity. The coordinates of the focus can be expressed as:
- \( Ae = 4 \) (since the focus is at (4,0))
- The directrix is given by \( x = \frac{A}{e} = 1 \).

### Step 4: Solve for \( A \) and \( e \)
From the equations:
1. \( Ae = 4 \)
2. \( \frac{A}{e} = 1 \)

From the second equation, we can express \( A \) in terms of \( e \):
\[ A = e \]

Now substitute \( A = e \) into the first equation:
\[ e^2 = 4 \]
Thus, \( e = 2 \).

### Step 5: Find the value of \( A \)
Now substituting \( e = 2 \) back into \( A = e \):
\[ A = 2 \]

### Step 6: Calculate \( B^2 \)
Using the relationship for eccentricity in hyperbolas:
\[ e^2 = 1 + \frac{B^2}{A^2} \]
Substituting the known values:
\[ 2^2 = 1 + \frac{B^2}{2^2} \]
\[ 4 = 1 + \frac{B^2}{4} \]
\[ 4 - 1 = \frac{B^2}{4} \]
\[ 3 = \frac{B^2}{4} \]
Thus, multiplying both sides by 4:
\[ B^2 = 12 \]

### Step 7: Write the equation of the hyperbola
The standard form of the hyperbola is given by:
\[ \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \]
Substituting \( A^2 = 4 \) and \( B^2 = 12 \):
\[ \frac{x^2}{4} - \frac{y^2}{12} = 1 \]

### Step 8: Clear the fractions
To eliminate the denominators, we can multiply through by 12:
\[ 3x^2 - y^2 = 12 \]

### Final Answer
The equation of the hyperbola is:
\[ 3x^2 - y^2 = 12 \]

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