Question

the derivation for the equation of a parabola with a vertex at the origin is started below.
\\(\sqrt{(x - 0)^2+(y - p)^2}=\sqrt{(x - x)^2+(y - (-p))^2}\\)

1. \\((x)^2+(y - p)^2=(0)^2+(y + p)^2\\)
2. \\(x^{2}+y^{2}-2py + p^{2}=y^{2}+2py + p^{2}\\)

if the equation is further simplified, which equation for a parabola does it become?
\\(x^{2}=4py\\)
\\(x^{2}=2y^{2}+2p^{2}\\)
\\(y^{2}=4px\\)
\\(y^{2}=4py\\)

Explanation:

Step1: Expand the equation

\[

$$\begin{align*} x^{2}+y^{2}-2py + p^{2}&=y^{2}+2py + p^{2}\\ x^{2}+y^{2}-2py + p^{2}-y^{2}-p^{2}&=y^{2}+2py + p^{2}-y^{2}-p^{2}\\ x^{2}-2py&=2py\\ x^{2}&=4py \end{align*}$$

\]

Answer:

$x^{2}=4py$