Finding equation of parabola with focus and directrix

Quote:

Given directrix y = -x + 2 and focus (0,0), find the equation of the parabola

So I found the equation for the perpendicular line to the directrix in order to find the vertex, which I got the line y = x that is perpendicular to the directrix, then solved the system of equations to find the common intersection point, which was (1,1). I used the midpoint formula from the focus to the directrix, and got (0.5,0.5) (even though I know by intuition). So the vertex is at (0.5,0.5) since it is exactly in between the focus and directrix.

Using the form , I plugged in the vertex already. To find the value of 'p' I needed to use the distance formula for between either the focus and vertex, or the vertex and directrix which I know should be exactly the same. So I did:

So:

Put in standard form:

I don't know if I've made any careless errors and I didn't want to rationalize the denominator and risk messing this up, but when I try to graph this, the graph of the parabola passes through the directrix...I thought it wasn't supposed to touch it at all?

Re: Finding equation of parabola with focus and directrix

Re: Finding equation of parabola with focus and directrix

That certainly makes a lot more sense...if I use that equation for every single parabola, including the ones with vertical/horizontal axes of symmetry, will I obtain the same result as if I used the other equation (in my original post)?

Re: Finding equation of parabola with focus and directrix

Quote:

Originally Posted by

**daigo** That certainly makes a lot more sense...if I use that equation for every single parabola, including the ones with vertical/horizontal axes of symmetry, will I obtain the same result as if I used the other equation (in my original post)?

You should get the same result yes. It just might involve a little more work.