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MathidiaMathematics Academy
Coordinate geometry and circles
Lesson 13 of 75
Standard practice

Equation of a circle

Objective. Interpret centre-radius form and complete the square.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

A circle with centre (a,b) and radius r has equation (x−a)²+(y−b)²=r², derived directly from the distance formula. If a circle equation is given expanded, such as x²+y²−4x+6y−3=0, completing the square on the x and y terms separately reveals the centre and radius: (x−2)²+(y+3)²=16, giving centre (2,−3) and radius 4. Always confirm r²>0, since a negative value means no real circle exists. This method connects algebraic manipulation directly to geometric interpretation.

Worked example

Find the centre and radius of the circle with equation x²+y²+6x−2y+6=0.

Diagram of a circle
  1. 1Group terms: (x²+6x) + (y²−2y) = −6
  2. 2Complete the square: (x+3)²−9 + (y−1)²−1 = −6
  3. 3Simplify to (x+3)² + (y−1)² = 4
  4. 4Read off centre (−3,1) and radius 2

Answer: Centre (−3, 1), radius 2

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Common mistake. Students read off the centre with wrong signs, taking (x−2)²+(y+3)²=16 to mean centre (−2,3) instead of (2,−3); the centre coordinates are the negatives of the constants inside each bracket.

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