Find, in standard form, the equation of : the parabola with focus at (3,5) and directrix at x=-1
Recall that the parabola
y^2 = 4px
has vertex at (0,0) and the distance from focus to directrix is 2p.
Your parabola's distance from focus to directrix is 4, so p=2. The focus is at (3,5), so the vertex halfway between the focus and directrix, at (1,5). So, your parabola is
(y-5)^2 = 8(x-1)
which agrees with Reiny's result.
To find the equation of a parabola given its focus and directrix, we can use the definition of a parabola.
The standard form of a parabola equation is given by (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix.
In this case, the vertex(h, k) is the midpoint between the focus and the directrix.
1. Find the vertex:
The x-coordinate of the vertex is the average of the x-coordinates of the focus and the directrix.
x-coordinate of vertex = (3 + (-1)) / 2 = 2 / 2 = 1
So, the x-coordinate of the vertex is 1.
The y-coordinate of the vertex remains the same as the y-coordinate of the focus since the parabola is vertical.
y-coordinate of vertex = 5
Therefore, the vertex is located at (1, 5).
2. Find the distance from the vertex to the focus or directrix:
Since the directrix is a vertical line x = -1, the distance from the directrix to the vertex is the absolute value of the difference of their x-coordinates.
Distance from vertex to directrix = |1 - (-1)| = |1 + 1| = 2
The distance from the vertex to the focus is the same as the distance from the vertex to the directrix, since both are measured along the y-axis.
So, the distance(p) is 2.
3. Write the equation in standard form:
Using the values we found, we can write the equation of the parabola in standard form:
(x - h)^2 = 4p(y - k)
(x - 1)^2 = 4(2)(y - 5)
(x - 1)^2 = 8(y - 5)
Therefore, the equation of the parabola with focus at (3, 5) and directrix at x = -1 is (x - 1)^2 = 8(y - 5)
To find the equation of a parabola given the focus and directrix, you can 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 and p is the distance between the focus and vertex.
In this case, the vertex is midway between the focus and directrix along the x-axis. The x-coordinate of the vertex is the average of the x-coordinates of the focus and the directrix:
Vertex x-coordinate = (3 + (-1))/2 = 2/2 = 1
Since the directrix is a vertical line with equation x = -1, the vertex has an x-coordinate of 1 and a y-coordinate of 5.
So the vertex is (1, 5).
The distance between the focus and the vertex is the distance between (3, 5) and (1, 5), which is:
p = 3 - 1 = 2
Now we have the vertex (1, 5) and the value of p = 2.
Plugging these values into the standard form equation, we get:
(x - 1)^2 = 4(2)(y - 5)
Expanding the equation gives:
(x - 1)^2 = 8(y - 5)
This is the standard form equation of the parabola with the given focus and directrix.
I don't like to show you by just plugging into some formula, since that way you really don't learn much
So I will do it by using the actual definition of the parabola
let P(x,y) be any point on our parabola.
P must be equidistant from (3,5) and the line x = -1
so x - (-1) = √( (x-3)^2 + (y-5)^2)
square both sides, and expanding
x^2 + 2x + 1 = x^2 - 6x + 9 + y^2 - 10y + 25 **
8x = y^2 - 10y + 33
x = (1/8)(y^2 - 10y + 33)
suppose we had left ** as
x^2 + 2x + 1 = x^2 - 6x + 9 + (y-5)^2
8x = (y-5)^2 + 8
x = (1/8)(y-5)^2 + 1
check my arithmetic
you might also look at this video by the Khan Academy
The parabola has a vertical directrix, yours has a horizontal directrix, but he explains it well
https://www.khanacademy.org/math/algebra2/intro-to-conics-alg2/focus-and-directrix-of-a-parabola-alg2/v/equation-for-parabola-from-focus-and-directrix