Using the given information write the equation of parabolas:
The parabola has focus (0, 6) and vertex at the origin;
x^2 = 24y
Well, well, well, it seems we have a parabola that likes to make a big fuss at the focus (0, 6) and loves hanging out at the vertex at the origin. Let's give this parabola its very own equation.
Since the parabola is symmetric about the y-axis (since the vertex is at the origin), we can assume the equation will take the form of y = a(x - h)^2 + k, where (h, k) represents the vertex.
Since the vertex is at the origin (0, 0), we can simplify this to y = ax^2.
Now we just need to find the value of 'a' to determine the shape of our parabola. The value of 'a' determines whether the parabola opens upwards or downwards.
Since the parabola has a focus at (0, 6), we know that it is a vertically oriented parabola that opens upwards. In this case, 'a' is positive.
So, our equation is y = ax^2, where 'a' is a positive constant.
Now, I'll leave it up to you to determine the exact value of 'a'.
To find the equation of a parabola, you need to determine whether it opens upwards or downwards.
In this case, since the focus is at (0, 6) and the vertex is at (0, 0), the parabola opens upwards because the focus is above the vertex.
The equation of a parabola with a vertical axis of symmetry that opens upwards can be written in the form:
(x - h)^2 = 4p(y - k)
Where (h, k) represents the coordinates of the vertex and p represents the distance from the vertex to the focus.
In this case, the vertex is at (h, k) = (0, 0) and the distance from the vertex to the focus is given as p = 6.
Therefore, substituting these values into the equation, we have:
(x - 0)^2 = 4(6)(y - 0)
Simplifying further:
x^2 = 24y
So, the equation of the parabola is x^2 = 24y.
recall that
x^2 = 4py has
vertex at (0,0)
focus at (0,p)
So, you have x^2 = 3/2 y