Choose the correct equation for the parabola with focus (0.5, 1) and directrix x=-0.5

The equation for a parabola can be found using the formula (x-h)² = 4p(y-k), where (h,k) is the vertex of the parabola and p is the distance from the vertex to the focus or the directrix.

Step 1: Find the vertex The vertex of the parabola is the midpoint between the focus and the directrix. In this case, the focus is at (0.5, 1) and the directrix is the line x = -0.5. The x-coordinate of the vertex is the average of the x-coordinates of the focus and the directrix, and the y-coordinate of the vertex is the same as the y-coordinate of the focus. So the vertex is ((0.5 - 0.5) / 2, 1) = (0, 1).

Step 2: Find the value of p The value of p is the distance from the vertex to the focus or the directrix. In this case, the distance from the vertex (0, 1) to the focus (0.5, 1) is 0.5. So p = 0.5.

Step 3: Substitute h, k, and p into the formula Substituting h = 0, k = 1, and p = 0.5 into the formula (x-h)² = 4p(y-k) gives (x - 0)² = 4 * 0.5 * (y - 1), which simplifies to x² = 2(y - 1).

So the correct equation for the parabola with focus (0.5, 1) and directrix x = -0.5 is x² = 2(y - 1).