The equation of the parabola with vertex (1,3) and focus (1,-1) is (x - 1)^2 = -8(y - 3).
A parabola is defined by its vertex and focus. The standard form equation of a parabola with a vertical axis is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus.
Given:
Vertex: (1,3)
Focus: (1,-1)
The x-coordinate of the vertex and the focus are the same, indicating that the parabola is symmetric about the y-axis. Thus, the equation will have the form (x - 1)^2 = 4p(y - 3).
To find the value of p, we need to calculate the distance between the vertex and the focus. The distance formula between two points (x1, y1) and (x2, y2) is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distance between (1,3) and (1,-1):
Distance = √((1 - 1)^2 + (-1 - 3)^2) = √(0 + 16) = √16 = 4
Since p is the distance from the vertex to the focus, we have p = 4. Substituting this into the equation, we get:
(x - 1)^2 = 4(4)(y - 3)
(x - 1)^2 = -16(y - 3)
(x - 1)^2 = -16y + 48
(x - 1)^2 + 16y = 48
(x - 1)^2 = -16y + 48
Therefore, the equation of the parabola with vertex (1,3) and focus (1,-1) is (x - 1)^2 = -16y + 48.
The equation of the parabola with vertex (1,3) and focus (1,-1) is (x - 1)^2 = -16y + 48.