Question

The directrix of a parabola is y=1. Its focus is (5,5).

what is the equation of the parabola in vertex form?

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y= (x- )^2 .

every space is a fill in the box.

Answer 1

The equation of the parabola with a directrix y=1 and focus at (5,5) is y = (1/8)(x - 5)^2 + 3.

Step-by-step explanation:

To find the equation of the parabola in vertex form with a directrix y=1 and a focus at (5,5), we first need to determine the vertex. The vertex lies midway between the focus and directrix, so it will have the same x-coordinate as the focus and a y-coordinate that is the average of the directrix y-value and the focus y-value.

For this parabola, the vertex is (5, 3). The parabola opens upward since the focus is above the directrix, and its equation takes the form y = a(x - h)^2 + k, where (h, k) is the vertex.

The value of a can be found by knowing that the distance from the vertex to the focus (p) is equal to 1/(4a). In this case, p = 2, so a = 1/8. The complete equation of the parabola is then y = (1/8)(x - 5)^2 + 3.