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The text defines the equation of a parabola as: $\sqrt{x^2+(y-p)^2}=y+p$
where $y$ is the y coordinate of a point on the parabola and $p$ is the y coordinate of the focus.

It goes on to say:

By squaring and simplifying we get $x^2=4py$.

I'm trying to recreate the steps they took to get from the first form to the second. I start by removing the radical sign by multiplying both sides by $\sqrt{x^2+(y-p)^2}$

but that doesn't seem to lead anywhere useful. What am I overlooking?

Thanks in advance.

Nick
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1 Answers1

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So, we have $\sqrt{x^2+(y-p)^2} = y + p$. If we square both sides, then we get $x^2 + (y-p)^2 = (y+p)^2$, whence $x^2 = (y+p)^2 - (y-p)^2 = 4py$.

Christopher K
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  • Ah...my mistake, I think, was distributing the negative sign in $-(y-p)^2$ such that I had $x^2 = (y+p)^2 + (-y+p)(-y+p)$, when in fact it should be $x^2 = (y+p)^2 + (-y+p)(y+p)$. But why don't we distribute the negative to the 2nd factor, $(y-p)$, also? – Nick Jun 05 '14 at 02:47