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Question 329448: Can you show me how to get the equation of a parabola with a focus (17,9) and a directrix x=12?
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Can you show me how to get the equation of
a parabola with a focus (17,9) and a directrix x=12.
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Draw the picture of the line and and the point.
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The vertex is half way between at (19.5, 9)
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p is the distance from the vertex to the focus = 2.5
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Form: (y-k)^2 = 4p(x-h)
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(y-9)^2 = 8(x-(39/2))
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Cheers,
Stan H.
Answer by Edwin McCravy(20059)  (Show Source):
You can put this solution on YOUR website! Can you show me how to get the equation of a parabola with a focus (17,9) and a directrix x=12?
The other tutor got the vertex wrong.
Let's draw the directrix line and the vertex:
The vertex is halfway between the focus and the directrix.
We draw a green line from the focus to the directrix
That green line is 5 units long, so the midpoint of the green line,
which is the vertex, is  units from the directrix and the
focus. So the coordinates of the vertex is ( ,9) or
( ,9)
To sketch in the parabola we construct two squares, one on each side of
the green line from vertex to focus:
Now we can sketch in the parabola with the vertex and which passes
through the corners of those two squares:
The equation of the parabola which opens right or left and has vertex (h,k)
is given by:
  
where p is the or  unit distance between the
directrix and the vertex, and also the same distance from the vertex
to the focus. p is taken positive if the parabola opens right, and
negative if the parabola opens left.
This parabola opens right so  , and with the vertex
(h,k) = (,9)
 
That's the equation of the parabola in standard form.
Edwin
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