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A parabola can be drawn given a focus of (3, 9)(3,9) and a directrix of y=1y=1. Write the equation of the parabola in any form.

User Sway
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1 Answer

9 votes
9 votes

Answer:

y = 1/16(x -3)² +5

Explanation:

The equation of a parabola can be written ...

y = 1/(4p)(x -h)² +k

where the vertex is (h, k), and p is the distance from the focus to the vertex. The vertex is halfway between the focus and directrix.

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Here, the focus has y-value 9, and the directrix has y-value 1. The vertex will have y-value (9+1)/2 = 5. The focus is above the directrix, so the parabola opens upward. The focus is on the line of symmetry, as is the vertex, so the vertex coordinates are (3, 5). The focus-vertex distance is ...

p = 9-5 = 4

Using the above form, we have the equation ...

y = 1/(4·4)(x -3)² +5

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Additional comment

Attached is a graph. The latus rectum is the horizontal line segment through the focus between one side of the parabola and the other. The ends of it are equidistant from the focus and the directrix. (That is also true of every other point on the parabola.)

A parabola can be drawn given a focus of (3, 9)(3,9) and a directrix of y=1y=1. Write-example-1
User Grebulon
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