1) A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2. Then the length of the semi-major axis is

For an ellipse with eccentricity e=1/2, focus at origin, and directrix x=4, the semi-major axis a satisfies a = e * distance from center to directrix. The center is at (-4/3, 0), so the directrix is at x = 4 (a/e to the right of center), giving a/e + a*e = 4, or a*(1/(1/2) + 1/2) = 4. Solving: (2a + a/2) = 4, so 5a/2 = 4, and a = 8/3. The ellipse equation x²/(64/9) + y²/(16) = 1 confirms this.