Find the equation of the parabola with the vertex at the origin and directrix y =5
Why did the parabola bring an umbrella to the origin? Because it heard the directrix was raining on its parade!
The equation of the parabola with the vertex at the origin and the directrix y = 5 can be written as y = 1/4x^2. Keep in mind that this equation is valid assuming the parabola is facing upwards. If it's facing downwards, just multiply the equation by -1 to flip it.
To find the equation of a parabola with a vertex at the origin and a directrix at y = 5, we need to determine the focus point and the equation that represents the shape of the parabola.
Step 1: Determine the focus point.
The focus point is equidistant from the vertex and the directrix. Since the vertex is at the origin (0, 0), the focus point will also lie on the y-axis. The distance between the vertex and the directrix is 5 units. Therefore, the focus point will be at (0, -5).
Step 2: Determine the equation of the parabola.
The general equation for a parabola with a vertical axis of symmetry is:
(x - h)² = 4p(y - k)
Where (h, k) represents the vertex, and p represents the distance between the vertex and the focus.
In this case, the vertex is at (0, 0), and the focus is at (0, -5). Therefore, the equation becomes:
x² = 4p(y - 0)
Simplifying further, we have:
x² = 4py
Since the vertex is at the origin, the equation becomes:
x² = 4py
Thus, the equation of the parabola with the vertex at the origin and the directrix y = 5 is x² = 4py.
To find the equation of a parabola with a vertex at the origin and a directrix at y = 5, we can use the standard form equation of a parabola, which is:
((x - h)^2) = 4p(y - k)
where (h, k) represents the vertex of the parabola and p is the distance from the vertex to the focus and from the vertex to the directrix.
In this case, the vertex is at the origin (0, 0), so h = 0 and k = 0.
The directrix is y = 5, which means the distance from the vertex to the directrix is 5 units. Since the vertex is at the origin, the distance from the origin to the directrix is the same as the distance from the origin to the focus.
Therefore, p = 5.
Now, we can substitute these values into the standard form equation to get the equation of the parabola:
(x - 0)^2 = 4(5)(y - 0)
Simplifying this equation gives us:
x^2 = 20y
So, the equation of the parabola with the vertex at the origin and directrix y = 5 is x^2 = 20y.
The vertex is mid-way between the directrix and focal point:
D(0,5)
V(0,0)
F(0,y)
FV = VD = 1/4a = 5,
0 - y = 5,
Y = -5.
1/4a = 5,
Multiply both sides by 4a:
20a = 1,
a = 1/20.
Vertex Form: Y = a(x-h)^2 + k,
Y = 1/20(x-0)^2 + 0,
Eq: Y =(1/20)x^2