Directrix at y=3,axis x=2 Latus rectum 4
assuming you mean a parabola,
recall that x^2 = 4py has
axis x=0
directrix at y = -p
latus rectum = 4p
So, your equation is
(x-2)^2 = 4(y-4)
see
http://www.wolframalpha.com/input/?i=parabola+(x-2)%5E2+%3D+4(y-4)
To find the equation of the parabola given the directrix, axis, and latus rectum, you can follow these steps:
1. Start by understanding some key concepts:
- The directrix is a horizontal line, y = k (where k is a constant).
- The axis of the parabola is a vertical line, x = a (where a is a constant).
- The latus rectum is the line segment passing through the focus and perpendicular to the axis of the parabola. Its length is twice the focal length.
2. Since the directrix is y = 3, we know that the parabola is concave downward and symmetric with respect to the line y = 3.
3. The axis of the parabola is x = 2, which means the parabola is symmetric with respect to the line x = 2.
4. The latus rectum has a length of 4, which means the distance between the vertex and either of its endpoints is equal to 2.
5. Using the vertex form of the parabola equation, we have:
(x - h)^2 = 4p(y - k)
h and k are the coordinates of the vertex, and p is the distance between the vertex and the focus.
6. Since the parabola is symmetric with respect to x = 2, the vertex must have the form (2, k).
7. Since the parabola is symmetric with respect to y = 3, the focus must have the form (2, k + p), where p is the distance between the vertex and the focus.
8. Since the latus rectum is 4 units long, the focus must be either 2 units above or below the vertex. Let's assume it is 2 units above the vertex: focus = (2, k + 2).
9. The distance between the vertex and the directrix is equal to p. In this case, it is 3 - k.
10. Since the parabola is symmetric, we have two possible focal lengths. Let's assume the focal length is positive, so p = 3 - k.
11. Now we can substitute the values into the equation from step 5:
(x - 2)^2 = 4(3 - k)(y - 3)
Simplifying the equation gives the final equation of the parabola:
(x - 2)^2 = 4(3 - k)(y - 3)
So, the equation of the parabola with the given directrix at y = 3, axis x = 2, and latus rectum 4 is:
(x - 2)^2 = 4(3 - k)(y - 3)