# Thread: Proof of a hyperbola centered at (h,k)

1. ## Proof of a hyperbola centered at (h,k)

I need to prove the standard formula of a hyperbola centered at (h,k). I can complete the proof for a hyperbola centered at (0,0) with no trouble, but I have had no success when the center is moved to (h,k). I am not sure if I am missing some substitution along the way or if my algebra is just wrong or what but it never comes out correctly. Any help or insight regarding the methods you would use or the steps you would follow would be greatly appreciated. Thank you.

2. Originally Posted by Jnorman223
I need to prove the standard formula of a hyperbola centered at (h,k). I can complete the proof for a hyperbola centered at (0,0) with no trouble, but I have had no success when the center is moved to (h,k). I am not sure if I am missing some substitution along the way or if my algebra is just wrong or what but it never comes out correctly. Any help or insight regarding the methods you would use or the steps you would follow would be greatly appreciated. Thank you.
Use the concept of translation: x --> x - h, y --> y - k in your equation for a hyperbola centred at (0, 0) .....

3. That is what I have done but somewhere along the way i am missing something or taking a wrong step.

My most recent dead end was at
$a^2=(x-h)^2+2c(x+h)+c^2+(y-k)^2$

I started with
$(sqrt(x-(h-c))^2+(y-k)^2))-(sqrt(x-(h+c))^2+(y-k)^2))=2a$

sorry i dont have any idea how to make it display a square root sign for the two distances on here but it should make sense.

### proof of eqn of a hyperbola centred (x, h)

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