Let's continue from where we left off to further simplify the given equation for the parabola:
Given:
[tex]\[ \sqrt{(x-0)^2 + (y-p)^2} = \sqrt{(x-x)^2 + (y-(-p))^2} \][/tex]
Simplify under the square root:
[tex]\[ (x)^2 + (y-p)^2 = (0)^2 + (y+p)^2 \][/tex]
Expand both sides:
[tex]\[ x^2 + y^2 - 2py + p^2 = y^2 + 2py + p^2 \][/tex]
Subtract [tex]\(y^2 + p^2\)[/tex] from both sides:
[tex]\[ x^2 - 2py = 2py \][/tex]
Combine like terms:
[tex]\[ x^2 = 4py \][/tex]
Thus, the equation is simplified to:
[tex]\[ x^2 = 4py \][/tex]
Therefore, the correct equation corresponding to the parabola in the given question is:
[tex]\[ x^2 = 4py \][/tex]
