Write an equation for the parabola with focus (0, 0) and directrix y = 1.
I got -x^2 - 2y + 1, but how do I get it in the form y - k = 1/4p(x - h)^2?
My equation was :
x^2 = 2y + 1
(Yours in not even an equation.)
manipulating this:
y = (1/2)(x^2) - 1/2
from this I can read that the vertex is (0,-1/2) which is true according to the focus - directrix property in that the vertex is midway between the focus and the directrix.
or
y + 1/2 = (1/2)(x - 0)^2
But that equation has a directrix of y = -1.
To find the equation of a parabola given its focus and directrix, we need to use the definition of a parabola.
Let's start by understanding the basic form of a parabola equation:
(y - k) = 1/4p(x - h)^2
In this form, (h, k) represents the vertex of the parabola, and p is the distance from the directrix to the focus.
Given that the focus is (0, 0) and the directrix is y = 1, we can determine the value of p, the distance between the focus and directrix.
The distance formula between a point (x, y) and a line y = mx + b is given by:
Distance = |y - mx - b| / sqrt(1 + m^2)
We can use this formula to find the distance between the point (0, 0) and the line y = 1:
Distance = |0 - 1(0) - 1| / sqrt(1 + 1^2) = 1 / sqrt(2) = sqrt(2) / 2
Now that we have the value of p, we can substitute it into the equation. In this case, p = sqrt(2) / 2.
Substituting the values of h, k, and p into the equation, we get:
(y - 1) = (1 / (4(sqrt(2) / 2)))(x - 0)^2
Simplifying further:
(y - 1) = (1 / (4(sqrt(2) / 2)))(x^2)
= (1 / (4(sqrt(2) / 2)))(x^2)
= (1 / (4 / sqrt(2)))(x^2)
= (sqrt(2) / 4)(x^2)
= (x^2 / 4sqrt(2))
Thus, the equation of the parabola in the form y - k = 1/4p(x - h)^2 is:
y - 1 = (x^2 / 4sqrt(2))