What defines the equation of a circle?

What defines the equation of a circle?

• A. a(x-h)² + b(y-k)² = r², with a = b
• B. ax² + by² = r²
• C. ax(x-h) + by(y-k) = r
• D. ab(x-h) + cd(y-k) = r²

Answer: (A)

The equation of a circle is most accurately defined by the formula that expresses the relationship between the coordinates of points on the circle and its center and radius. In this case, a circle centered at the point (h, k) with a radius r can be represented in the standard form as (x - h)² + (y - k)² = r². The choice that specifies a(x-h)² + b(y-k)² = r², with a = b, corresponds to this standard definition because when a = b, the equation simplifies to (x - h)² + (y - k)² = r². This holds true for any circle, ensuring that all points (x, y) that satisfy this equation are exactly those lying on the circle's circumference. The equality of a and b is crucial here, as it maintains the symmetrical nature of a circle. The other options do not accurately denote the geometric property of a circle. For example, more complex forms or different proportions between terms fail to encapsulate the essential characteristics of a circle's standard definition as derived from its center and radius.

The equation of a circle is most accurately defined by the formula that expresses the relationship between the coordinates of points on the circle and its center and radius. In this case, a circle centered at the point (h, k) with a radius r can be represented in the standard form as (x - h)² + (y - k)² = r².

The choice that specifies a(x-h)² + b(y-k)² = r², with a = b, corresponds to this standard definition because when a = b, the equation simplifies to (x - h)² + (y - k)² = r². This holds true for any circle, ensuring that all points (x, y) that satisfy this equation are exactly those lying on the circle's circumference. The equality of a and b is crucial here, as it maintains the symmetrical nature of a circle.

The other options do not accurately denote the geometric property of a circle. For example, more complex forms or different proportions between terms fail to encapsulate the essential characteristics of a circle's standard definition as derived from its center and radius.