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I'm trying to solve this problem to study for a competition:

The endpoints of the latus rectum (focal diameter) of a parabola are (-1, 2) and (7, 2). The vertex of this parabola lies in the first quadrant. Determine the coordinates of this vertex. Express your answer as the ordered pair (x, y)

So what I've figured out so far is that the focus is (3, 2) just because it's the midpoint of the latus rectum. Also, this means that the x-coordinate of the vertex is 3. I just can't figure out how to find the y-coordinate of the vertex without more information.

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  • $\begingroup$ The distance from the focus to a point on the parabola (say $(7,2)$) equals the distance from the point to the directrix. Now you know the directrix... $\endgroup$ – B. Goddard Feb 23 '18 at 20:22
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    $\begingroup$ Half the latus rectum equals the distance from the focus to the directrix. The fact that the vertex is in the first quadrant should tell you whether the vertex is above or below the latus rectum. $\endgroup$ – Fabio Somenzi Feb 23 '18 at 20:24
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Please verify that ( vertex-focus distance)= $\frac12$ length of semi-latus-rectum, as a property for all parabolas.

The vertex is above or below $ 4/2=2 $ units from $(y=2)$ latus-rectum horizontal line. So y-coordinate is either 0 or 4.

Coordinates $(3,0),(3,4) $ are two possibilities for vertex position.

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