Final answer:
The equation of the parabola with vertex at (0,0) and directrix at x= -1/2 is x = (1/2)y^2. This is derived from setting the squared distances from a point on the parabola to the focus and directrix equal to each other.
Step-by-step explanation:
The question asks for the equation of a parabola with its vertex at the origin (0,0) and the directrix at x = -1/2. To find the equation of the parabola, we need to understand that a parabola is the set of all points that are equidistant from a fixed point called the focus and a line called the directrix. The distance from the vertex to the focus and from the vertex to the directrix is the same, and in this case, the distance is 1/2, since the directrix is at x = -1/2.
Since the vertex is at the origin and the directrix is vertical, we know that the parabola must open to the right. This means the parabola's equation will be of the form x=ay^2. The focus will be at (1/2, 0), which is on the positive x-axis 1/2 units away from the vertex. Each point on the parabola is an equal distance from the focus as it is from the directrix. The distance from the focus to a point (x, y) on the parabola is the square root of ((x-1/2)^2 + y^2). The distance from the point (x, y) to the directrix is |x + 1/2|. Setting these two distances equal gives us the equation of the parabola.
From the definition of a parabola, we can set the squared distances equal to each other without square roots: (x - 1/2)^2 + y^2 = (x + 1/2)^2. This simplifies to x - 1/2 + y^2 = x + 1/2 or y^2 = 2x. Writing in the required format we get x = (1/2)y^2.