Find the equation of the parabola whose directrix is y = -1 and focus is (1, 1).
To find the equation of a parabola given its directrix and focus, we need to use the standard form of the equation of a parabola:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex of the parabola, p is the distance from the vertex to either the focus or the directrix, and the parabola opens upward or downward depending on whether p is positive or negative.
In this case, the directrix is y = -1 and the focus is (1, 1).
Step 1: Determine the vertex:
Since the directrix is a horizontal line (y = -1), the vertex lies on the perpendicular line passing through the midpoint of the segment between the focus and the directrix.
The midpoint of the segment connecting the focus (1, 1) and the directrix (any point with y = -1) is (1, 0.5).
So the vertex is (1, 0.5).
Step 2: Determine the distance from the vertex to either the focus or the directrix:
The distance from the vertex to the directrix (y = -1) is 0.5 - (-1) = 1.5.
So p = 1.5.
Step 3: Substitute the known values into the standard equation to get the final equation:
(x - h)^2 = 4p(y - k)
(x - 1)^2 = 4(1.5)(y - 0.5)
(x - 1)^2 = 6(y - 0.5)
Thus, the equation of the parabola with the given directrix (y = -1) and focus (1, 1) is (x - 1)^2 = 6(y - 0.5).