The locus of the center of a circle that touches another circle externally and also touches the x-axis is represented by the equation of an upward-opening parabola: x² = 4y for y ≥ 0, along with all points (0, y) for y < 0. Therefore the correct choice is option a).
The problem given involves finding the locus of the center of a circle that touches another circle externally and also touches the x-axis. Considering the circle (y - 1)² + x² = 1, it has a center at (0,1) and a radius 1. A circle with center (h,k) that touches this circle externally will have a radius that equals the distance between (h,k) and (0,1) minus 1. This circle also touches the x-axis, meaning the y-coordinate of its center, k, equals its radius.
Following this logic, the equation is (h - 0)² + (k - 1)² = (k - 1 + 1)² which simplifies to h² + (k - 1)² = k². Since k is the radius, and it lies on the x-axis, it is always non-negative (k ≥ 0). Therefore, solving for k in terms of h, we get k = h²/4, valid for k ≥ 0. For k < 0, the circle would be beneath the x-axis and therefore cannot touch the x-axis.
The correct option for the locus of the center of such a circle is (a) {(x² = 4y, y ≥ 0), (0, y), y < 0}. The equation x² = 4y represents the upper half of the parabola that opens upwards, with (0, y) accommodating the lower half beneath the x-axis, which makes the entire parabola the locus of the circle's center.
