Question

The equation of a parabola is given. y=-1/12x^2-2x-1

What are coordinates of the focus?

Answer 1

`-(x^2+24x+144)=12y+12...`Answer:

(-12, 8)

Explanation:

The standard form of this parabola, the one we can use to determine the vertex coordinates and the value of p is:

`(x-h)^2=4p(y-k)`

where h and k are the coordinates of the vertex and p is the distance between the vertex and the focus. We need that p value to determine how far above the vertex the focus is. In this case, the focus will lie on the same x-coordinate as the focus, we just need to find how far that distance away is. That requires us to do some algebraic gymnastics on that original equation. Putting it into vertex form.

Begin by multiplying everything by 12 to get rid of the pesky fraction:

`12y=-x^2-24x-12`

Now we need to complete the square. The easiest way to do this is to have just the x terms on one side of the equals sign and everything else on the other side, so we will add 12 to both sides:

`-x^2-24x=12y+12`

The leading coefficient when you complete the square has to be a positive 1; ours is a negative 1, so factor out the negative:

`-(x^2+24x)=12y+12`

The rules for completing the square are as follows: Take half the linear term (ours is a 24), square that half, then add it into the parenthesis.

Half of 24 is 12 so

`-(x^2+24x+144)=12y+12`

BUT...since this is an equation, if we add something to one side we have to add it to the other side too. BUT we didn't just add in a 144, we have to take into account the -1 sitting outside the parenthesis that will not be ignored. So we didn't add in 144, we added in -1(144) which is -144.

`-(x^2+24x+144)=12y+12-144`

What we have done on the left by completing the square is to create a perfect square binomial. Rewriting it as such and combining like terms on the right:

`-(x+12)^2=12y-132`

Don't forget the purpose of this is to find the value of p. We're almost there. On the right, factor out a 12:

`-(x+12)^2=12(y-11)`

From this we can determine the coordinates of the vertex and the value of p. The vertex sits at (-12, 11).

The equation for p is 4p = 12 so p = 3

That means that the focus is 3 units below the vertex on the same x coordinate. The focus then is at (-12, 8)