What is the equation of the parabola with a focus at (-3,4) and a directrix of y= -7?

1 Answer
Douglas K.
Jul 8, 2017

#y = 1/22(x- (-3))^2 -3/2#

Explanation:

Because the directrix is a horizontal line, we know that the vertex has the same x coordinate as the focus and has a y coordinate that is the midpoint between the directrix and the focus:

#(h,k) = (-3,(4+ (-7))/2)#

#(h,k) = (-3,-3/2)#

Substitute the values for h and k into the vertex form for the equation of a parabola that opens vertically:

#y = a(x-h)^2+k#

#y = a(x- (-3))^2 -3/2#

Compute the signed vertical distance, #f#, from the vertex to the focus:

#f = 4 - (-3/2)#

#f = 11/2#

We know that #a = 1/(4f)#

#a = 1/(4(11/2))#

#a = 2/(4(11))#

#a = 1/22#

Substitute the value for "a" into the equation:

#y = 1/22(x- (-3))^2 -3/2#

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