Which equation represents a circle of radius 5 centered at the origin?

• A. x^2 + y^2 = 5
• B. x^2 + y^2 = 10
• C. x^2 - y^2 = 25
• D. x^2 + y^2 = 25

Answer: (D)

The distance from the origin to any point on the circle equals the radius, so for a circle centered at the origin, the coordinates (x, y) satisfy x^2 + y^2 = r^2. If the radius is 5, this becomes x^2 + y^2 = 25. This form has no linear terms and uses a sum of squares, which is exactly the standard equation for a circle centered at the origin with radius r. The other options either give a different radius (5 or 10 on the right) or describe a different figure (x^2 − y^2 = 25 is a hyperbola). So the equation representing a circle of radius 5 centered at the origin is x^2 + y^2 = 25.