What is the equation of the parabola with vertex (3,4) and focus (6,4)?
AAAaannndd the bot gets it wrong yet again!
Sorry, but (x - 3)^2 = 12(y - 4)
has vertex = (3,4) and focus = (3,7)
Since the vertex is on the same horizontal line (y=4) as the focus and to the left of the focus, the axis is horizontal, and the parabola opens to the right.
Since the focus is 3 units to the right of the vertex, p = 3, so
(y-4)^2 = 12(x-3)
no, the directrix is the line 3 units to the left of the vertex: x = 0
I apologize for my mistake. You are correct that the directrix is a vertical line 3 units to the left of the vertex, which is x = 0. Thank you for correcting me and please let me know if you have any further questions.
Since the vertex is (3,4), the equation of the parabola must be of the form:
(x - 3)^2 = 4p(y - 4)
where p is the distance from the vertex to the focus. We know that the focus is (6,4), so p = 3.
Substituting p = 3 into the equation above, we get:
(x - 3)^2 = 4(3)(y - 4)
Simplifying, we have:
(x - 3)^2 = 12(y - 4)
This is the equation of the parabola.
is this right x=12(y-4)^2+3
No, the equation x = 12(y - 4)^2 + 3 is not correct for the given vertex and focus.
The correct form of the equation of a parabola with vertex (h, k) and focus (h, k + p) (or (h + p, k) for a horizontally oriented parabola) is:
(x - h)^2 = 4p(y - k)
In this case, the vertex is (3, 4) and the focus is (6, 4), so the value of p is 3.
Therefore, the equation of the parabola is:
(x - 3)^2 = 4(3)(y - 4)
Simplifying, we get:
(x - 3)^2 = 12(y - 4)
This is the correct equation of the parabola.
I apologize for my previous error. You are correct, the equation (x - 3)^2 = 12(y - 4) is the equation of the parabola with vertex (3,4) and focus (3,7).
The axis of symmetry is the vertical line passing through the vertex, which is x = 3. Since the parabola opens to the right and p = 3, the distance between the vertex and the directrix is also 3 units. Thus, the equation of the directrix is y = 1.
Thank you for bringing this to my attention and please let me know if you have any further questions.