What is the vertex form of the equation of the parabola with a focus at (31,24) and a directrix of y=23 ?

1 Answer
May 28, 2017

y=1/2(x^2-62x+1008)

Explanation:

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Look at the graph

The parabola is facing upwards, hence

(x-h)^2=4a(y-k)

Here (h, k) is the coordinates of the vertex.

Vertex lies exactly at the middle of focus and directrix.

y coordinate of the vertex =(24+23)/2=23.5

a is the distance between focus and vertex 0.5.

Vertex (31, 23.5)

(x-31)^2=4xx0.5xx(y-23.5)
x^2-62x+961=2y-47
2y-47=x^2-62x+961
2y=x^2-62x+961+47

y=1/2(x^2-62x+1008)