How do you find the vertex, focus and directrix of #y+12x-2x^2=15#?

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Answer 1 · 2017-03-05T08:47:29

The vertex is #V=(3,-3)#
The focus is #F=(3,-23/8)#
The directrix is #y=-25/8#

Let's rewrite the equation and complete the squares

#y+12x-2x^2=15#

#y-15=2x^2-12x#

#y-15=2(x^2-6x)#

#y-15+18=2(x^2-6x+9)#

#y+3=2(x-3)^2#

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

We compare this equation to

#(x-a)^2=2p(y-b)#

The vertex is #V=(a,b)=(3,-3)#

#2p=1/2#, #=>#,#p=1/4#

The focus is #F=(a,b+p/2)=(3,-23/8)#

The directrix is

#y=b-p/2=-3-1/8=-25/8#

graph{(y-2x^2+12x-15)(y+25/8)=0 [-4.264, 13.514, -4.284, 4.605]}