What is the vertex form of the equation of the parabola with a focus at (31,24) and a directrix of #y=23 #?

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Answer 1 · 2017-05-28T05:03:23

#y=1/2(x^2-62x+1008)#

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Look at the graph

The parabola is facing upwards, hence

#(x-h)^2=4a(y-k)#

Here #(h, k) # is the coordinates of the vertex.

Vertex lies exactly at the middle of focus and directrix.

y coordinate of the vertex #=(24+23)/2=23.5#

#a# is the distance between focus and vertex #0.5#.

Vertex #(31, 23.5)#

#(x-31)^2=4xx0.5xx(y-23.5)#
#x^2-62x+961=2y-47#
#2y-47=x^2-62x+961#
#2y=x^2-62x+961+47#

#y=1/2(x^2-62x+1008)#