Answer :
To determine the standard equation for a circle centered at the origin with radius [tex]\( r \)[/tex], we need to recall the definition and properties of a circle in a coordinate plane.
The general equation for a circle centered at the origin [tex]\((0,0)\)[/tex] is derived based on the distance from the center of the circle to any point [tex]\((x, y)\)[/tex] on the circle. The equation utilizes the Pythagorean Theorem:
1. A circle consists of all points that are a fixed distance (the radius) away from the center.
2. For a circle centered at [tex]\((0,0)\)[/tex] with radius [tex]\( r \)[/tex], any point [tex]\((x,y)\)[/tex] on the circle satisfies the relationship:
[tex]\[ \sqrt{x^2 + y^2} = r \][/tex]
3. Squaring both sides of this equation to eliminate the square root gives:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
Given the options provided:
A. [tex]\( x^2 + y^2 = r \)[/tex]
B. [tex]\( x + y = r \)[/tex]
C. [tex]\( x^2 = y^2 + r^2 \)[/tex]
D. [tex]\( x^2 + y^2 = r^2 \)[/tex]
Among these, option D ([tex]\( x^2 + y^2 = r^2 \)[/tex]) correctly represents the standard equation of a circle centered at the origin with radius [tex]\( r \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
This aligns with option D.
The general equation for a circle centered at the origin [tex]\((0,0)\)[/tex] is derived based on the distance from the center of the circle to any point [tex]\((x, y)\)[/tex] on the circle. The equation utilizes the Pythagorean Theorem:
1. A circle consists of all points that are a fixed distance (the radius) away from the center.
2. For a circle centered at [tex]\((0,0)\)[/tex] with radius [tex]\( r \)[/tex], any point [tex]\((x,y)\)[/tex] on the circle satisfies the relationship:
[tex]\[ \sqrt{x^2 + y^2} = r \][/tex]
3. Squaring both sides of this equation to eliminate the square root gives:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
Given the options provided:
A. [tex]\( x^2 + y^2 = r \)[/tex]
B. [tex]\( x + y = r \)[/tex]
C. [tex]\( x^2 = y^2 + r^2 \)[/tex]
D. [tex]\( x^2 + y^2 = r^2 \)[/tex]
Among these, option D ([tex]\( x^2 + y^2 = r^2 \)[/tex]) correctly represents the standard equation of a circle centered at the origin with radius [tex]\( r \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
This aligns with option D.
