Find an equation for the parabola that has its vertex at the origin and directrix x=-9.
If your answer is in the form x^2=Ay, then input here: A=
if your answer is in the form y^2=Ax, then input here: A= .
using y^2 = 4px , p = 9
so we get y^2 = 36x , placing the focus on the x-axis
comparing this with your y^2 = Ax
would make A = 36
To find the equation for the parabola with vertex at the origin and directrix x = -9, we can start by using the definition of a parabola.
A parabola is the set of all points that are equidistant to the focus F and the directrix. In this case, since the vertex is at the origin, the focus F will be located at a point on the positive x-axis that is equidistant from the origin and the directrix x = -9.
Let's determine the coordinates of the focus F.
Since the vertex is at the origin, the distance between the origin and F is the same as the distance between the origin and the directrix x = -9. This distance is the perpendicular distance from the origin to the directrix, which is simply the absolute value of the x-coordinate of the directrix.
So, the distance between the origin and F is |(-9)| = 9 units.
Since the focus is located on the positive x-axis, its coordinates are (9, 0).
Now, let's use the standard form of a parabola equation, which is given by:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance between the vertex and the focus.
In our case, the vertex is (0, 0) and the focus is (9, 0), so we have:
(x - 0)^2 = 4p(y - 0)
x^2 = 4py
The value of p is the distance between the vertex and the focus, which is 9 units.
Plugging in this value, we get:
x^2 = 4(9)y
x^2 = 36y
So, the equation of the parabola is x^2 = 36y.
Therefore, in the form x^2 = Ay, A = 36.