What is an equation of a parabola with the given vertex and focus?
vertex: (–2,5); focus: (–2,6)
Since the focus and vertex of a parabola lie on the axis of symmetry, the equation of the parabola can be written in the form:
(x – h)^2 = 4p(y – k)
where (h,k) is the vertex of the parabola and p is the distance between the vertex and the focus.
In this case, the vertex is (-2,5) and the focus is (-2,6). Therefore, the distance between the vertex and the focus is p = 1.
Substitute the values into the equation:
(x + 2)^2 = 4(1)(y - 5)
Simplifying:
(x + 2)^2 = 4y - 20
Expand the square:
x^2 + 4x + 4 = 4y - 20
Rearrange the terms to get the final equation:
x^2 + 4x - 4y + 24 = 0
Therefore, the equation of the parabola with the given vertex and focus is x^2 + 4x - 4y + 24 = 0.