Fill in the blank(s) to correctly complete each sentence. The circle with equation $x^2+y^2=49$ has center with coordinates________ and radius equal to__________ .

Verified step by step guidance

1. Recall the standard form of a circle's equation: \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
2. Compare the given equation \(x^2 + y^2 = 49\) to the standard form. Notice that it can be rewritten as \((x - 0)^2 + (y - 0)^2 = 7^2\).
3. From this comparison, identify the center coordinates \((h, k)\) as \((0, 0)\) because there are no terms shifting \(x\) or \(y\).
4. Identify the radius \(r\) by taking the square root of 49, which is \(7\).
5. Therefore, the circle has center at \((0, 0)\) and radius equal to \(7\).

Key Concepts

Standard Form of a Circle Equation

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

Center of a Circle

The center of a circle is the point (h, k) from which all points on the circle are equidistant. In the equation x^2 + y^2 = 49, the center is at the origin (0, 0) because there are no (x - h) or (y - k) terms.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. It is found by taking the square root of the constant on the right side of the equation r^2. For x^2 + y^2 = 49, the radius is √49 = 7.