What are the focus and directrix of the parabola represented by the equation:8(y-6)=(x+2)squared?
You know that the standard parabola
x^2 = 4py
has focus at (0,p) and directrix at y = -p.
Your parabola has been shifted so its vertex is at (-2,6). 4p=8.
So, the focus is 2 above the vertex and the directrix is 2 below the vertex.
To find the focus and directrix of the parabola represented by the equation 8(y-6)=(x+2)^2, we can compare it to the standard form of a parabola equation, which is (x-h)^2=4p(y-k), where (h,k) is the vertex and p is the distance from the vertex to the focus and directrix.
First, let's rewrite the given equation in standard form:
8(y-6)=(x+2)^2
Divide both sides by 8:
(y-6)=(1/8)(x+2)^2
Comparing it to the standard form, we have:
(x-h)^2=4p(y-k)
In our equation,
h = -2
k = 6
4p = 1/8
From 4p = 1/8, divide both sides by 4:
p = (1/8)/4
p = 1/32
Therefore, the vertex is (-2, 6) and p = 1/32.
To find the focus, we can add p to the y-coordinate of the vertex:
(6 + 1/32) = 193/32
So, the focus is (-2, 193/32).
Now, let's find the directrix. Since the parabola opens upward, the directrix is a horizontal line below the vertex, and its equation is y = k - p.
Substituting the known values:
y = (6 - 1/32)
y = (191/32)
Therefore, the directrix is the line y = 191/32.
In summary,
The focus of the parabola is (-2, 193/32), and the directrix is the line y = 191/32.
To find the focus and directrix of a parabola represented by an equation, we need to rewrite the equation in a standard form called the vertex form. The vertex form of a parabola equation is given by:
(y - k) = 4a(x - h)^2
Where (h, k) represents the coordinates of the vertex of the parabola, and a determines the shape and size of the parabola.
Given equation: 8(y - 6) = (x + 2)^2
We can rearrange the equation to match the vertex form:
8(y - 6) = (x + 2)^2
Divide both sides by 8:
(y - 6) = (1/8)(x + 2)^2
This equation is now in the vertex form.
Comparing this with the standard vertex form, we can see that the vertex of this parabola is at (-2, 6).
To find the value of a, which determines the shape and size of the parabola, we recognize that 4a = 1/8.
Dividing both sides by 4:
a = (1/8)/4
a = 1/32
Now we have the vertex (-2, 6) and a value of a = 1/32.
To find the focus and directrix, we can use the following formulas:
Focus: (h + 1/(4a), k)
Directrix: y = k - 1/(4a)
Substituting the values we found:
Focus: (-2 + 1/(4 * 1/32), 6)
Simplifying:
(-2 + 1/8, 6)
(-15/8, 6)
Therefore, the focus of the given parabola is (-15/8, 6).
Directrix: y = 6 - 1/(4 * 1/32)
Simplifying:
y = 6 - 8
y = -2
Therefore, the directrix of the given parabola is y = -2.
To summarize, the focus of the parabola represented by the equation 8(y-6) = (x+2)^2 is (-15/8, 6), and the directrix is y = -2.