What is the equation for the parabola with vertex (0, 0) and focus (5, 0)
I am not sure if you just want to write the equation, or if you actually want to develop the equation.
I will assume the latter.
if focus is (5,0) and vertex is (0,0), the the directrix is x = -5
Let P(x,y) be any point on the parabola
x+5 = √( (x-5)^2 + y^2)
square both sides
x^2 + 10x + 25 = x^2 - 10x + 25 + y^2
20x = y^2 is your equation
To find the equation for a parabola given its vertex and focus, you can use the standard form equation for a parabola:
(x - h)^2 = 4p(y - k)
where (h, k) represents the vertex of the parabola, and p represents the distance between the vertex and the focus.
In this case, the vertex is at (0, 0), so h = 0 and k = 0. The focus is at (5, 0), which means the distance between the vertex and the focus, p, is 5 units.
Plugging these values into the equation, we have:
(x - 0)^2 = 4(5)(y - 0)
Simplifying further:
x^2 = 20y
Therefore, the equation for the parabola with vertex (0, 0) and focus (5, 0) is x^2 = 20y.