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. Identify the standard form of a circle's equation, which is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
2. Compare the given equation \( x^2 + y^2 = 49 \) with the standard form.
3. Notice that the equation can be rewritten as \( (x - 0)^2 + (y - 0)^2 = 49 \), indicating that \(h = 0\) and \(k = 0\).
4. Determine the center of the circle from the values of \(h\) and \(k\), which are both 0, so the center is \((0, 0)\).
5. Find the radius by taking the square root of 49, which is \(r = \sqrt{49}\).

Key Concepts

Circle Equation

The standard equation of a circle in a Cartesian coordinate system is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. In this case, the equation x² + y² = 49 can be rewritten to identify the center and radius directly.

Center of a Circle

The center of a circle is the point from which all points on the circle are equidistant. For the equation x² + y² = 49, the center is at the origin (0, 0) because there are no h or k values subtracted from x or y, indicating that the circle is centered at the coordinate axes.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. In the equation x² + y² = 49, the radius can be found by taking the square root of 49, which equals 7. Thus, the radius is 7 units.