Use the equation of a circle centred at the origin

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Use the equation of a circle centred at the origin

A point on a circle of radius r centred at the origin has horizontal and vertical distances whose squares sum to r². The equation is x² + y² = r². It comes from a right triangle with the radius as hypotenuse.

The radius is the square root of the right-hand constant, not the constant itself. For x² + y² = 25, r = 5. The axis intersections are (±5, 0) and (0, ±5), and the circle has symmetry in both coordinate axes.

Substitution distinguishes points on, inside and outside the circle: compare x² + y² with r². This comparison assumes the usual equal-unit Cartesian geometry; a distorted display can make the circle look oval without changing the equation.

Worked example: Check whether (3, 4) lies on x² + y² = 25.

Square the coordinates separately.

Add 9 + 16.

Compare with 25.

Answer: Yes: the sum is 25.

Check your understanding: What is the radius of x² + y² = 49?

Answer: 7.

Why: The equation contains the radius squared.