General Certificate of Secondary Education (GCSE) Maths Practice Exam

The equation that represents the relationship of a circle with a center at the origin, (0,0), is given by the formula x² + y² = r². In this equation, r represents the radius of the circle. This equation describes all the points (x, y) that are a fixed distance 'r' from the center (0,0).

When you square the x and y coordinates and add them together, you are calculating the distance from the origin to any point on the circle. If this sum equals r², it confirms that the point lies on the circumference of the circle, which is why this equation effectively defines a circle.

The other options represent different mathematical concepts:

- The line equation y = mx + b describes a straight line with slope m and y-intercept b.

- The equation y = k^x represents an exponential function, which shows rapid growth or decay rather than a circular relationship.

- The equation y = x² describes a parabola that opens upwards, which does not have the properties of a circle.

Thus, the equation x² + y² = r² is specifically tailored to demonstrate the geometric relationship of a circle centered at the origin.