SOLUTION: Write the equation of the parabola that has its focus at the point (4,7) and whose directrix is x=-2.

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Question 975216: Write the equation of the parabola that has its focus at the point (4,7) and whose directrix is x=-2.
Answer by Edwin McCravy(20059)   (Show Source): You can put this solution on YOUR website!

The trouble with parabolas is that different books use different notation.
They are all equivalent, but they are just different enough to cause confusion.
I will arbitrarily pick one of the notations.  If your book is different, just
tell me what your book gives for the standard equation for a parabola in the
thank-you note below and I'll change it on here to fit your notation. One
common one is:



where (h,k) = the vertex, p is the distance from center to focus, positive
if parabola opens right and negative if it opens left.

We plot the focus point and draw the the directrix, a vertical line
through x=-2 on the x-axis:



The vertex is a point exactly half-way between the focus and the directrix
line. That is the point (h,k) = (1,7)



p = the distance from the center to the focus is 3 units here.

The focal chord (sometimes called the latus rectum) is a line which is 4p
units long (in this case 4*3 or 12 units long) bisected at the focus.  So
it's 6 units up from the focus and 6 units down from the focus:


  
And now we can sketch the parabola:



h=1,k=7, 4p = 12, so the equation is



Edwin

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