To solve the problem, we need to find the coordinates of the center of the ellipse given its focus, directrix, and eccentricity.
### Step-by-step Solution:
1. **Identify the Given Information:**
- Focus \( F(3, 4) \)
- Directrix \( x + y - 1 = 0 \)
- Eccentricity \( e = \frac{1}{2} \)
2. **Equation of the Directrix:**
- The directrix can be rewritten in the standard form \( Ax + By + C = 0 \) where \( A = 1, B = 1, C = -1 \).
3. **Find the Perpendicular Line (Axis of the Ellipse):**
- The slope of the directrix is \( -1 \) (from \( y = -x + 1 \)).
- The slope of the line perpendicular to this (the axis of the ellipse) will be \( 1 \).
- Thus, the equation of the axis can be expressed as \( y - 4 = 1(x - 3) \) or \( y = x + 1 \).
4. **Substituting the Focus into the Axis Equation:**
- Substitute \( F(3, 4) \) into the axis equation to find \( k \):
\[
4 = 3 + k \implies k = 1
\]
- Therefore, the equation of the axis is \( y = x + 1 \).
5. **Finding the Foot of the Directrix:**
- The foot of the directrix can be found by substituting \( y = x + 1 \) into the directrix equation:
\[
x + (x + 1) - 1 = 0 \implies 2x = 0 \implies x = 0
\]
- Substituting \( x = 0 \) back into \( y = x + 1 \):
\[
y = 0 + 1 = 1
\]
- Thus, the foot of the directrix \( D(0, 1) \).
6. **Using the Section Formula to Find the Center:**
- The center \( C \) divides the segment \( FD \) in the ratio \( 1:3 \) (since \( e = \frac{1}{2} \) implies \( 1: (2e) = 1:1 \)).
- Using the section formula:
\[
C\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\]
- Here, \( F(3, 4) \) is \( (x_1, y_1) \) and \( D(0, 1) \) is \( (x_2, y_2) \) with \( m = 1 \) and \( n = 3 \):
\[
C_x = \frac{1 \cdot 0 + 3 \cdot 3}{1 + 3} = \frac{9}{4} = 2.25
\]
\[
C_y = \frac{1 \cdot 1 + 3 \cdot 4}{1 + 3} = \frac{13}{4} = 3.25
\]
7. **Final Coordinates of the Center:**
- The coordinates of the center \( C \) are \( \left( \frac{9}{4}, \frac{13}{4} \right) \).
### Final Answer:
The coordinates of the center of the ellipse are \( \left( 2.25, 3.25 \right) \).
