parabola equations

adkinsjr · #1 $Old$ July 4th, 2009, 11:03 PM

I am want to derive some of the equations for parabolas with vertex at (h,k). I wrote the equation below directly from the definition. I'm having a hard time obtaining any of the standard forms for parabolas from this definition. For example: y = a(x-h)^2 +K is one such equation I would like to derive. I need to know if I'm setting this up right.

$p=\sqrt{(x-h)^2+[y-(k+p)]^2}$

$p^2=(x-h)^2+y^2-2y(k+p) + (k+p)^2$

$(x-h)^2=-y^2+2yk-k^2+2p(y-k)$

$(x-h)^2=-(y-k)^2+2p(y-k)$

As you can see I'm having trouble converting this into either form:

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

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

Where a =1/4p

malaygoel · #2 $Old$ July 4th, 2009, 11:07 PM

Quote:

Originally Posted by adkinsjr $View Post$

I am want to derive some of the equations for parabolas with vertex at (h,k). I wrote the equation below directly from the definition. I'm having a hard time obtaining any of the standard forms for parabolas from this definition. For example: y = a(x-h)^2 +K is one such equation I would like to derive. I need to know if I'm setting this up right.

What is P here?

distance of (x,y) from (h,k+p)
= $\sqrt{(x-h)^2+(y-k-p)^2}$

distance of (x,y) from line y=k-p
= $y-k+p$

Can you proceed now?

adkinsjr · #3 $Old$ July 4th, 2009, 11:27 PM

Quote:

Originally Posted by malaygoel $View Post$

What is P here?

distance of (x,y) from (h,k+p)
= $\sqrt{(x-h)^2+(y-k-p)^2}$

distance of (x,y) from line y=k-p
= $y-k+p$

Can you proceed now?

P is the distance from (x,y) to the line y = k - P

I should be able to proceed from here. I just needed to know if I was setting this up right. I didn't want to waste time with the algebra if I didn't have the rigth equation.

malaygoel · #4 $Old$ July 5th, 2009, 01:17 AM

Quote:

Originally Posted by adkinsjr $View Post$

P is the distance from (x,y) to the line y = k - P

P = y - k + p

I should be able to proceed from here. I just needed to know if I was setting this up right. I didn't want to waste time with the algebra if I didn't have the rigth equation.

yeah, you are setting up the right equation.