atticus30
Answered

The derivation for the equation of a parabola with a vertex at the origin is started below. If the equation is further simplified, which equation for a parabola does it become?

[tex]\[
\sqrt{(x-0)^2+(y-p)^2}=\sqrt{(x-0)^2+(y+p)^2}
\][/tex]

1. [tex]\((x)^2+(y-p)^2=(0)^2+(y+p)^2\)[/tex]

2. [tex]\(x^2+y^2-2py+p^2=y^2+2py+p^2\)[/tex]

Which equation does it simplify to?

A. [tex]\(x^2 = 4py\)[/tex]

B. [tex]\(x^2 = 2y^2 + 2p^2\)[/tex]

C. [tex]\(y^2 = 4px\)[/tex]

D. [tex]\(y^2 = 4py\)[/tex]



Answer :

GinnyAnswer
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]

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