To find the equation of a parabola given its vertex and focus, we start with the basic definition of a parabola and its properties.
The vertex of the parabola is given as (2, 5), and the focus, which is a point that lies on the axis of symmetry of the parabola, is given as (2, 6). Since the x-coordinates of the vertex and focus are the same, we can infer that the parabola opens upwards.
The standard form of the equation of a vertically oriented parabola can be expressed as:
(x – h)² = 4p(y – k)
In this equation, (h, k) is the vertex of the parabola, and 4p represents the distance from the vertex to the focus. Here, (h, k) = (2, 5).
Next, we find the distance p. Since the focus (2, 6) is one unit above the vertex (2, 5), we have p = 1. This means that 4p = 4.
Now we can substitute these values into the standard form:
(x – 2)² = 4(1)(y – 5)
Therefore, the equation simplifies to:
(x – 2)² = 4(y – 5)
This is the required equation of the parabola with vertex (2, 5) and focus (2, 6).