Question

How do you find the vertex, focus, and directrix of the parabola `y^2+6y+8x+25=0`?

Answer 1

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

Explanation:

Rewrite the equation and complete the squares

`y^2+6y+8x+25=0`

`y^2+6y=-8x-25`

`y^2+6y+9=-8x-25+9`

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

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

We compare this equation to

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

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

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

The focus is `(a+p/2,b)=(-4,-3)`

The directrix is `x=a-p/2`, `=>`, `x=-2+2=0`

graph{(y^2+6y+8x+25)((x+2)^2+(y+3)^2-0.1)((x+4)^2+(y+3)^2-0.1)=0 [-59.95, 13.1, -21.5, 15.05]}