CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The circle with equation x² + y² = 49 has center with coordinates ________ and radius equal to _______.

Verified step by step guidance

1. Recognize that the given equation is in the form of a circle equation: \(x^{2} + y^{2} = r^{2}\), where the center is at the origin \((0,0)\) and \(r\) is the radius.
2. Identify the center coordinates by comparing the equation to the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\). Here, \(h = 0\) and \(k = 0\), so the center is \((0,0)\).
3. Determine the radius by taking the square root of the constant on the right side of the equation. Since the equation is \(x^{2} + y^{2} = 49\), the radius \(r = \sqrt{49}\).
4. Express the radius as \(r = 7\) (without calculating the final value, just the expression to find it).
5. Summarize: The center is at \((0,0)\) and the radius is \(7\).

Key Concepts

Standard Equation of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) represents the center coordinates and r is the radius. Recognizing this form helps identify the circle's center and radius directly from the equation.

Center of a Circle

The center of a circle is the fixed point equidistant from all points on the circle. In the equation (x - h)² + (y - k)² = r², the center is at (h, k). If the equation is x² + y² = r², the center is at the origin (0, 0).

Radius of a Circle

The radius is the distance from the center to any point on the circle. It is the square root of the constant term on the right side of the equation (r²). For example, if the equation is x² + y² = 49, the radius is √49 = 7.