Let's analyze and simplify the given equation step-by-step to determine which standard form of the equation of a parabola it becomes.
Given the equation:
[tex]\[ \sqrt{(x-0)^2 + (y-p)^2} = \sqrt{(x-x)^2 + (y-(-p))^2} \][/tex]
First, we can simplify it:
[tex]\[ \sqrt{x^2 + (y-p)^2} = \sqrt{(y+p)^2} \][/tex]
Simplifying the right-hand side:
[tex]\[ \sqrt{x^2 + (y-p)^2} = |y+p| \][/tex]
We need to consider the absolute value because it represents the distance, which is always positive. The general understanding here will follow:
[tex]\[ \sqrt{x^2 + (y-p)^2} = |y+p| \][/tex]
Now let's square both sides to eliminate the square roots:
[tex]\[ x^2 + (y-p)^2 = (y+p)^2 \][/tex]
Expanding both sides:
[tex]\[ x^2 + y^2 - 2py + p^2 = y^2 + 2py + p^2 \][/tex]
Subtracting [tex]\(y^2 + p^2\)[/tex] from both sides, we get:
[tex]\[ x^2 - 2py = 2py \][/tex]
Combining like terms:
[tex]\[ x^2 - 4py = 0 \][/tex]
Thus we have the final simplified equation:
[tex]\[ x^2 = 4py \][/tex]
Therefore, the equation of the parabola [tex]\( \sqrt{(x-0)^2 + (y-p)^2} = \sqrt{(x-x)^2 + (y-(-p))^2} \)[/tex] simplifies to [tex]\( x^2 = 4py \)[/tex].
So, the correct option is:
[tex]\[ x^2 = 4py \][/tex]
