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Sagot :
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.
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.
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