Circle Equation from Centre and Radius

Quick answer: A circle with centre (h, k) and radius r > 0 has equation (x − h)² + (y − k)² = r². It describes points whose distance from the centre is r.

Worked example

Write the equation for centre (2, −3) and radius 5.

1. Substitute h = 2, k = −3, r = 5.
2. The standard form is (x − 2)² + (y + 3)² = 25.
3. Expand if required: x² − 4x + 4 + y² + 6y + 9 = 25.
4. Hence x² + y² − 4x + 6y − 12 = 0.

Check the result

Point (7, −3) is 5 units right of the centre; substitution gives 25 + 0 = 25.

Common mistake to avoid

The signs inside the brackets are opposite to the centre coordinates. The right side is radius squared, not radius.

Try it yourself

1. Identify centre and radius of (x + 1)² + (y − 4)² = 9. 2. Write a circle of radius 2 centred at the origin.

Answers and reasoning

1. Centre (−1, 4), radius 3. 2. x² + y² = 4.

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