help please!! write an equation of a parabola with vertex at the origin and the given focus
focus at (2,6)??? Please help im lost!!
Recall that the parabola x^2=4py has
vertex at (0,0)
focus at (0,p)
Clearly your parabola has been rotated through an angle θ where tanθ = 6/2
The distance from (0,0) to (2,6) is √40=2√10.
So, in the rotated coordinate system (x',y') we have
y' = 8√10 x'^2
Now, if you haven't yet studied rotation of axes, then you must have a typo. Otherwise, if the vertex is at (0,0) then the focus must be on one of the axes.
it says the focus is at (2,6) and vertex is at origin. We covered a little bit about rotation of axes
Oh, well, in that case, we can proceed. I misspoke. The way I rotated the axes, I should have said
x' = 8√10 y'^2
I'd rather have y=ax^2, so I should have said
x'^2 = 8√10 y'
or
y' = 1/(8√10) x'^2
I'll get back to you on how that works with the rotation of axes. Since the focus is at (2,6) it means that the axes have been rotated clockwise through an angle θ such that tanθ = 2/6.
So,
sinθ = -1/√10
cosθ = 3/√10
You can use those values in your rotation matrix.
To write the equation of a parabola with a given focus, you need to know the vertex and focus. In this case, the focus is given as (2, 6), and the vertex is mentioned to be at the origin (0, 0).
The general equation of a parabola in standard form with a vertical axis is given by:
(x - h)^2 = 4p(y - k)
where (h, k) represents the vertex, and p represents the distance between the vertex and the focus.
Since the vertex is at the origin (0, 0), we have h = 0 and k = 0.
Substituting these values into the equation, we have:
x^2 = 4py
Now, to find the value of p, we can use the distance formula:
p = distance from vertex to focus
Using the coordinates of the vertex (0, 0) and the focus (2, 6), the distance formula becomes:
p = sqrt((2 - 0)^2 + (6 - 0)^2)
= sqrt(4 + 36)
= sqrt(40)
= 2√10
Substituting this value of p into the equation, we get:
x^2 = 4(2√10)y
Therefore, the equation of the parabola with the given focus at (2, 6) and the vertex at the origin is:
x^2 = 8√10y