To find the equation of a parabola, we need to determine a few key elements: the vertex, the focus, and the direction it opens. In this case, we have the following information:
- Focus: (5, 3)
- Vertex: (5, 6)
The vertex is located at (5, 6), and the focus is at (5, 3). Since the focus is below the vertex, we know that the parabola opens downwards.
The general form of the equation of a parabola that opens downward is given by:
(x – h)2 = -4p(y – k)
where:
- (h, k) is the vertex of the parabola
- p is the distance from the vertex to the focus
In our case:
- The vertex (h, k) is (5, 6).
- To determine p, we calculate the distance from the vertex to the focus:
Distance, p = k – focus_y = 6 – 3 = 3
Since the parabola opens downward, p will be negative, thus:
p = -3
Now substituting h, k, and p into the standard parabola equation:
(x – 5)2 = -4(-3)(y – 6)
This simplifies to:
(x – 5)2 = 12(y – 6)
So the final equation of the parabola is:
(x – 5)2 = 12(y – 6)
In summary: The equation of the parabola with focus at (5, 3) and vertex at (5, 6) is:
(x – 5)2 = 12(y – 6)