Question
Find the equation of a parabola with focus at (3, 2) and directrix at y = 8.
Answers
GPT 3.5
Since the focus is at (3, 2) and the directrix is y = 8, the parabola is vertical.
The vertex of the parabola is the midpoint of the segment connecting the focus and the directrix. The midpoint is at (3, 5).
The distance between the vertex and the focus is the same as the distance between the vertex and the directrix. This distance is 3 units.
The equation of a vertical parabola with vertex at (h, k), focus at (h, k + p), and directrix y = k - p is given by:
(x - h)^2 = 4p(y - k)
Substitute the values we have:
(x - 3)^2 = 4(3)(y - 5)
(x - 3)^2 = 12(y - 5)
(x - 3)^2 = 12y - 60
x^2 - 6x + 9 = 12y - 60
12y = x^2- 6x + 69
y = (1/12)x^2 - (1/2)x + 23/4
So, the equation of the parabola with focus at (3, 2) and directrix at y = 8 is y = (1/12)x^2 - (1/2)x + 23/4.
The vertex of the parabola is the midpoint of the segment connecting the focus and the directrix. The midpoint is at (3, 5).
The distance between the vertex and the focus is the same as the distance between the vertex and the directrix. This distance is 3 units.
The equation of a vertical parabola with vertex at (h, k), focus at (h, k + p), and directrix y = k - p is given by:
(x - h)^2 = 4p(y - k)
Substitute the values we have:
(x - 3)^2 = 4(3)(y - 5)
(x - 3)^2 = 12(y - 5)
(x - 3)^2 = 12y - 60
x^2 - 6x + 9 = 12y - 60
12y = x^2- 6x + 69
y = (1/12)x^2 - (1/2)x + 23/4
So, the equation of the parabola with focus at (3, 2) and directrix at y = 8 is y = (1/12)x^2 - (1/2)x + 23/4.