Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.