Locus and Conic Definitions of Conic Sections

I need the conic and locus definition of a hyperbola and the conic definition of a parabola. All the definitions or part of one is asked for.

Thank you so much!

All right. I haven't done this since Algebra and Pre-Calc, so I hope I remembered right.

So for a hyperbola, I believe it is

(x-h)^2 - (y-k)^2 = 1
/a^2........./b^2

(Ignore ...... It is to fill space)

This is where h,k is center.
The (x-h) term would be positive if horizontally major. (open sideways)
The (y-k) term would be positive if vertically major (up and down)

Locus, I think you mean focus?

So take the square root of a^2 + b^2.
Then add that to (h,k)
but it depends on the orientation, so you add it to h or k depending on if it is major one way or the other.


Parabola was
y=a(x-h)^2+k

Where h,k is vertex.

You also have x=a(y-k)^2 + h
if you want it sideways.


To find the focus or directrix of a parabola:

Take the absolute value of 1/(4a).

Then add that to (h,k)
and again it depends if concave up or down
or if sideways.