Question

How do you find the vertex, focus and directrix of `y^2 + 6y + 8x - 7 = 0`?

Answer 1

The vertex is `V=(2,-3)`
The focus is `F=(0,-3)`
The directrix is `x=4`

Explanation:

Let's rearrage the equation and complete the squares

`y^2+6y+8x-7=0`

`y^2+6y=-8x+7`

`y^2+6y+9=-8x+7+9`

`(y+3)^2=-8x+16=-8(x-2)`

We compare this equation to

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

`2p=-8`, `=>`, `p=-4`

The vertex is `V=(a,b)=(2,-3)`

The focus is `F=(a+p/2,b)=(0,-3)`

The equation of the directrix is

`x=a-p/2=2+2=4`

graph{(y^2+6y+8x-7)(y+100x-400)((x)^2+(y+3)^2-0.01)=0 [-10, 10, -5, 5]}