What is the vertex form of the equation of the parabola with a focus at (15,-5) and a directrix of #y=7 #?

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Answer 1 · 2015-11-28T02:17:00

Conic form: #(x-11)^2 = 16(y-5)#

We know this up/down parabola #=># equation of directrix is #y=7#

General equation #(x-h)^2 =4p(y-k)#

Focus #(h+p, k)#

Directrix: #y= h-p#

Focus #(15, -5)# so
1) #h+p= 15# ; #color(red)(k= -5)#

Directrix: #y= 7#
2) #h-p=7#

Now we have system of equation (add both equations)
# h+p= 15#
#h--p = 7#
#=> 2h =22 =># #color(red)(h= 11)#

Now we solve for #p#
#h+p= 15#
#(11)+p =15# #=> color(red)(p= 4)#

Conic form
#(x-color(red)11)^2 =4*color(red)(4)(y-color(red)5)#
Answer: #(x-11)^2 = 16(y-5)#