The equation of the chord of the ellipse `x^(2) + 4y^(2) = 4` having the middle point at `(-2, (1)/(2))` is

Options

A. `2x - 2y + 7 = 0`
B. `x + 2y = 0`
C. `3x - 2y + 4 = 0`
D. `2x - 2y + 5 = 0`

Answer

To find the equation of the chord of the ellipse \( x^2 + 4y^2 = 4 \) that has its midpoint at the point \( (-2, \frac{1}{2}) \), we can use the formula for the equation of a chord with a given midpoint.

### Step-by-Step Solution:

1. **Identify the Ellipse Equation**:
The given ellipse is \( x^2 + 4y^2 = 4 \). We can rewrite it in standard form:
\[ \frac{x^2}{4} + \frac{y^2}{1} = 1 \]

2. **Identify the Midpoint**:
The midpoint of the chord is given as \( (-2, \frac{1}{2}) \). Let’s denote this point as \( (x_1, y_1) \) where \( x_1 = -2 \) and \( y_1 = \frac{1}{2} \).

3. **Use the Chord Formula**:
The equation of the chord with midpoint \( (x_1, y_1) \) is given by:
\[ T = S_1 \]
where \( T \) is the equation of the chord and \( S_1 \) is the value of the ellipse equation at the midpoint.

4. **Calculate \( S_1 \)**:
Substitute \( x_1 \) and \( y_1 \) into the ellipse equation:
\[ S_1 = x_1^2 + 4y_1^2 - 4 \]
\[ S_1 = (-2)^2 + 4\left(\frac{1}{2}\right)^2 - 4 \]
\[ S_1 = 4 + 4 \cdot \frac{1}{4} - 4 \]
\[ S_1 = 4 + 1 - 4 = 1 \]

5. **Set Up the Equation for \( T \)**:
The equation for \( T \) is:
\[ T = x x_1 + 4y y_1 - 4 \]
Substitute \( x_1 = -2 \) and \( y_1 = \frac{1}{2} \):
\[ T = x(-2) + 4y\left(\frac{1}{2}\right) - 4 \]
\[ T = -2x + 2y - 4 \]

6. **Set \( T \) Equal to \( S_1 \)**:
Now we set \( T = S_1 \):
\[ -2x + 2y - 4 = 1 \]

7. **Rearrange the Equation**:
Rearranging gives:
\[ -2x + 2y - 5 = 0 \]
or multiplying through by -1:
\[ 2x - 2y + 5 = 0 \]

### Final Answer:
The equation of the chord of the ellipse \( x^2 + 4y^2 = 4 \) having the midpoint at \( (-2, \frac{1}{2}) \) is:
\[ 2x - 2y + 5 = 0 \]