How do you find the vertex, focus and directrix of x^2 - 8x - 16y + 16 = 0x28x16y+16=0?

1 Answer
Dec 13, 2016

The vertex is =(4,0)=(4,0)
The focus is =(4,4)=(4,4)
The directrix is y=-4y=4

Explanation:

Let's rewrite the equation

x^2-8x+16=16yx28x+16=16y

16y=(x-4)^216y=(x4)2

y=1/16(x-4)^2y=116(x4)2

(x-4)^2=16y(x4)2=16y

This is the equation of a parabola.

(x-a)^2=2p(y-b)(xa)2=2p(yb)

Where a=4a=4, b=0b=0 and p=8p=8

The vertex is V=(a,b)=(4,0)V=(a,b)=(4,0)

The focus is F=(a,b+p/2)=(4,4)F=(a,b+p2)=(4,4)

And the directrix is y=b-p/2=0-8/2=-4y=bp2=082=4

graph{(16y-(x-4)^2)(y+4)=0 [-13.53, 22.5, -4.68, 13.34]}