What is the equation of the parabola, given the following equation:
Directrix is the y-axis and focus at (3,5)
The definition of a parabola:
The set of points which are equidistant from a given focus and a given directrix.
Let one such point be (x,y)
so √( (x-0)^2 + (y-y)^2 ) = √( (x-3)^2 + (y-5)^2)
square both sides and simplify
x^2 = x^2 - 6x + 9 + y^2 - 10y + 25
6x = y^2 - 10y + 34
x = (1/6)(y^2 - 10y + 34)
check my arithmetic,
Recall that the parabola y^2 = 4px has
focus at (p,0)
directrix x = -p
vertex at (0,0)
Note that the vertex is midway between the focus and the directrix.
So, your parabola has vertex at (3/2,5)
p = 3/2
So, (y - 5)^2 = 4(3/2)(x - 3/2)
as Reiny derived above.
I now get it. Thank you very much!
To find the equation of a parabola given the directrix and focus, we need to find the vertex and the value of the focal length. The vertex is the midpoint between the focus and the directrix, and the focal length is the distance between the vertex and the focus.
In this case, the directrix is the y-axis, which means it is a vertical line with the equation x = 0. The focus is given as (3, 5).
First, let's find the vertex. Since the directrix is the y-axis, the x-coordinate of the vertex is zero. The y-coordinate of the vertex is the average of the y-coordinate of the focus and the directrix, which in this case is (5 + 0) / 2 = 2.5. Therefore, the vertex is (0, 2.5).
Next, let's find the value of the focal length. The focal length is the distance between the vertex and the focus. Using the distance formula, we have:
√((3 - 0)^2 + (5 - 2.5)^2)
= √(9 + 2.25)
= √11.25
= 3√1.25
Now that we have the vertex (0, 2.5) and the focal length 3√1.25, we can write the equation of the parabola in vertex form as:
(x - h)^2 = 4a(y - k)
where (h, k) is the vertex and a is the focal length.
Substituting the values, we have:
(x - 0)^2 = 4(3√1.25)(y - 2.5)
Simplifying further:
x^2 = 12√1.25(y - 2.5)
Therefore, the equation of the parabola is x^2 = 12√1.25(y - 2.5).