What is the equation in standard form of the parabola with a focus at (10,-9) and a directrix of y= -14?

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Answer 1 · 2016-02-08T07:46:07

#y=x^2/10-2x-3/2#

from the given focus #(10, -9)# and equation of directrix #y=-14#,
calculate #p#

#p=1/2(-9--14)=5/2#

calculate the vertex #(h, k)#

# h=10# and #k=(-9+(-14))/2=-23/2#

Vertex #(h, k)=(10, -23/2)#

Use the vertex form
#(x-h)^2=+4p(y-k)# positive #4p# because it opens upward

#(x-10)^2=4*(5/2)(y--23/2)#

#(x-10)^2=10(y+23/2)#

#x^2-20x+100=10y+115#
#x^2-20x-15=10y#

#y=x^2/10-2x-3/2#

the graph of #y=x^2/10-2x-3/2# and the directrix #y=-14#
graph{(y-x^2/10+2x+3/2)(y+14)=0[-35,35,-25,10]}