Focus (4, 0), e =` 1/2` , directrix is x -16 = 0. Then equation of the ellipse is

Options

A. `(x^(2))/(16)+(y^(2))/(9)=1`
B. `(x^(2))/(64)+(y^(2))/(32)=1`
C. `(x^(2))/(64)+(y^(2))/(48)=1`
D. `(x^(48))/(16)+(y^(2))/(64)=1`

Answer

To find the equation of the ellipse given the focus, eccentricity, and directrix, we can follow these steps:

### Step 1: Identify the given values
- Focus: \( (4, 0) \)
- Eccentricity \( e = \frac{1}{2} \)
- Directrix: \( x - 16 = 0 \) or \( x = 16 \)

### Step 2: Determine the orientation of the ellipse
Since the focus is at \( (4, 0) \) and is on the x-axis, the major axis of the ellipse is horizontal. Thus, the standard form of the ellipse will be:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

### Step 3: Calculate the value of \( c \)
The distance from the center to the focus is denoted as \( c \). Since the focus is at \( (4, 0) \), we have:
\[ c = 4 \]

### Step 4: Relate \( a \), \( e \), and the directrix
The equation of the directrix for an ellipse is given by:
\[ x = \frac{a}{e} \]
Given that the directrix is \( x = 16 \), we can set up the equation:
\[ \frac{a}{e} = 16 \]
Substituting \( e = \frac{1}{2} \):
\[ \frac{a}{\frac{1}{2}} = 16 \implies a = 16 \cdot \frac{1}{2} = 8 \]

### Step 5: Calculate \( b \)
We know the relationship between \( a \), \( b \), and \( c \) for ellipses:
\[ b^2 = a^2 - c^2 \]
Substituting the values we found:
\[ a = 8 \implies a^2 = 64 \]
\[ c = 4 \implies c^2 = 16 \]
Now substituting these into the equation:
\[ b^2 = 64 - 16 = 48 \]

### Step 6: Write the equation of the ellipse
Now that we have \( a^2 \) and \( b^2 \), we can write the equation of the ellipse:
\[ \frac{x^2}{64} + \frac{y^2}{48} = 1 \]

### Final Answer
The equation of the ellipse is:
\[ \frac{x^2}{64} + \frac{y^2}{48} = 1 \]

---