Write an equation of a parabola with a vertex at the origin and a directrix at y=-5.
my answer is focus 6,0, directrix =-6
(x-0)^2 = 4a(y-0)
a = distance vertex to focus or to directrix
if a is +, parabola holds water
here directrix is below vertex, so a is +5
x^2 = 20 y
To find the equation of a parabola with a vertex at the origin (0,0) and a directrix at y = -5, we can use the formula:
(y - k)^2 = 4a(x - h)
In this case, since the vertex is at the origin (0,0), the values of h and k are both 0. Now let's find the value of a.
The distance from the vertex to the directrix is the same as the distance from the vertex to the focus. In this case, the directrix is y = -5, and the distance from the vertex to the directrix is 5 units.
Since the vertex is at the origin, the focus will be 5 units above the vertex, which means the focus is at (0, 5).
To find the value of a, which represents the distance from the vertex to the focus, we use the formula a = 1/(4p), where p is the distance between the vertex and the focus.
Since the distance between the vertex (0,0) and the focus (0,5) is 5 units, a = 1/(4 * 5) = 1/20.
Now we have all the necessary information to write the equation of the parabola:
(y - 0)^2 = 4(1/20)(x - 0)
Simplifying the equation further:
y^2 = (1/5)(x)
So, the equation of the parabola with a vertex at the origin and a directrix at y = -5 is y^2 = (1/5)(x).
To write the equation of a parabola with a vertex at the origin and a directrix at y=-5, we can use the standard form of the equation for a parabola:
(y - k)^2 = 4p(x - h)
where (h, k) is the vertex, and p is the distance from the vertex to the focus (and also the distance from the vertex to the directrix). In this case, since the vertex is at (0, 0), we have h = 0 and k = 0.
The directrix is given as y = -5, which means the distance from the vertex to the directrix is p = 5.
The focus, on the other hand, can be found as a point p units above the vertex, which means the focus is at (0, 5).
Substituting these values into the standard form equation gives us:
(y - 0)^2 = 4(5)(x - 0)
Simplifying this equation, we get:
y^2 = 20x
Therefore, the equation of the parabola with a vertex at the origin and a directrix at y = -5 is y^2 = 20x.