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?\\(\bigcirc\\ x^2 = 4py\\)\\(\bigcirc\\ x^2 = 2y^2 + 2p^2\\)\\(\bigcirc\\ y^2 = 4px\\)\\(\bigcirc\\ y^2 = 4py\\)

Explanation:

Step1: Subtract \(y^2 + p^2\) from both sides

We start with the equation \(x^2 + y^2 - 2py + p^2 = y^2 + 2py + p^2\). Subtracting \(y^2\) and \(p^2\) from both sides gives us \(x^2 - 2py = 2py\).

Step2: Add \(2py\) to both sides

Taking the equation \(x^2 - 2py = 2py\), we add \(2py\) to both sides. This results in \(x^2 = 2py + 2py\), which simplifies to \(x^2 = 4py\).

Answer:

\(x^2 = 4py\) (the first option)