Let's derive the correct standard equation for a circle centered at the origin with radius [tex]\( r \)[/tex].
1. Circle's General Equation: For a circle centered at the origin (0, 0) and with a radius [tex]\( r \)[/tex], the standard equation is derived from the Pythagorean Theorem.
2. Derivation Step-by-Step:
- If a point [tex]\((x, y)\)[/tex] lies on the circle, then the distance from this point to the origin should be equal to [tex]\( r \)[/tex].
- By the distance formula:
[tex]\[
\sqrt{x^2 + y^2} = r
\][/tex]
- To eliminate the square root, we square both sides of the equation:
[tex]\[
(\sqrt{x^2 + y^2})^2 = r^2
\][/tex]
- Simplifying, we get:
[tex]\[
x^2 + y^2 = r^2
\][/tex]
We now match this derived standard equation of the circle centered at the origin with the given choices:
A. [tex]\(x^2 + y^2 = r^2\)[/tex]
B. [tex]\(x^2 = y^2 + r^2\)[/tex]
C. [tex]\(x + y = r\)[/tex]
D. [tex]\(x^2 + y^2 = r\)[/tex]
Clearly, option A matches the derived equation perfectly:
[tex]\[
x^2 + y^2 = r^2
\][/tex]
Hence, the correct choice is A.
