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. Identify the general form of the equation of a circle, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
2. Compare the given equation x^2 + y^2 = 49 with the general form. Notice that the x and y terms do not have any coefficients or constants added or subtracted, which implies h = 0 and k = 0.
3. Conclude that the center of the circle, (h, k), is at the origin, which is (0, 0).
4. Identify the term on the right side of the equation, 49, which represents r^2, where r is the radius of the circle.
5. Determine the radius by taking the square root of 49, which gives the radius r.

Key Concepts

Equation of a Circle

The standard form of the equation of a circle is (x - h)² + (y - k)² = r², where (h, k) represents 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), as there are no h or k values subtracted from x or y in the standard form.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. It can be found by taking the square root of the value on the right side of the equation when in standard form. Here, since 49 = r², the radius is r = √49 = 7.