umm suppose a parabola has a vertex at (0,2) and points (1,1)
how would I derive the equation and focus, i've been trying to understand this for so long, I can't get it. Does this parabola have the equation (y-2)^2 = x and a focus of 1/4?? Is that correct??
I searched Google under the key words "parabola equation focus" to get these possible sources:
http://en.wikipedia.org/wiki/Parabola
http://www.tpub.com/math2/13.htm
http://colalg.math.csusb.edu/~devel/precalcdemo/conics/src/parabola.html
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_conics_directrix.xml
http://mathworld.wolfram.com/Parabola.html
Use the <Find> command to locate keywords within these sites.
I hope this helps. Thanks for asking.
Formats to help you find the equation for a parabola:
(x - h)^2 = 4p(y - k)
Vertex = (h, k)
Focus = (h, k + p)
Directrix: y = k - p
You are given the vertex (0,2) and a point (1,1).
Let's try to find p using the vertex and the point you were given:
(x - h)^2 = 4p(y - k)
(1 - 0)^2 = 4p(1 - 2)
1 = 4p(-1)
1 = -4p
-1/4 = p
Let's try to find the equation now that we know p:
(x - 0)^2 = 4(-1/4)(y - 2)
x^2 = -1(y - 2)
x^2 = -y + 2
Set the equation equal to 0:
x^2 + y - 2 = 0 -->this is the equation for the parabola.
(You can check the point you were given by substituting the values into the equation.)
Focus = (h, k + p)
Therefore: (0, 7/4) is the focus.
Directrix: y = k - p
Therefore: y = 9/4
In this case, the focus is below the directrix; therefore, the parabola opens downward and p is negative.
I hope this helps.
To derive the equation and focus of a parabola with a given vertex and point, you can follow these steps:
1. Use the general equation for a parabola in vertex form: (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus.
2. Plug in the given vertex coordinates (0, 2) into the equation: (x - 0)^2 = 4p(y - 2).
3. Next, substitute the coordinates of the given point (1, 1) into the equation to solve for p: (1 - 0)^2 = 4p(1 - 2).
4. Simplify the equation and solve for p: 1 = 4p(-1), 1 = -4p, p = -1/4.
5. Substitute the value of p into the equation: x^2 = -1(y - 2).
6. Set the equation equal to 0 to get the standard form: x^2 + y - 2 = 0. This is the equation for the parabola.
7. The focus is located at (h, k + p). Therefore, the focus is (0, 2 - 1/4) = (0, 7/4).
8. The directrix is given by the equation y = k - p. Therefore, the directrix is y = 2 - (-1/4) = 2 + 1/4 = y = 9/4.
In summary, the equation of the given parabola is x^2 + y - 2 = 0 and the focus is located at (0, 7/4), with the directrix given by y = 9/4.