Equation of Circle

1Summary 重點摘要

Learn how to write equation of circle — given centre and radius.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The standard formThe standard form of the circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r². Why this shape? It comes directly from the definition of a circle: every point on the circle lies at distance r from the centre, so the distance formula gives √((x − h)² + (y − k)²) = r, and squaring both sides yields the standard form. For centre (2, 3) and radius 5, the equation is (x − 2)² + (y − 3)² = 25. A common mistake is writing the radius instead of its square on the right, or getting the signs of the coordinates wrong.
2. Step 2: Writing the equation from centre and radiusGiven the centre and radius, substitute directly into the standard form. For centre (2, 3) and radius 5, the equation is (x − 2)² + (y − 3)² = 25; for centre (−1, 4) and radius 2, it is (x + 1)² + (y − 4)² = 4, where h = −1 turns x − h into x + 1. Why does a plus sign appear? Subtracting a negative number equals adding a positive one, and this sign change is the most error-prone spot. A common mistake is reading (x + 1)² as h = +1; remember the bracket is x minus the x-coordinate of the centre.
3. Step 3: The special case of centre at the originIf the centre is the origin (0, 0), then h = k = 0 and the standard form simplifies to x² + y² = r². For example, a circle of radius 4 has equation x² + y² = 16, and the unit circle has x² + y² = 1. Why list this case separately? It is the simplest form of the standard equation, allows the equation to be written at once, and provides a quick check of answers. A common mistake is writing x² + y² = 4 for radius 4, forgetting the right side is 4² = 16; radius and squared radius must be kept distinct.
4. Step 4: Expanding to the general formExpanding the standard form and collecting like terms gives the general form x² + y² + Dx + Ey + F = 0. Take (x − 2)² + (y − 3)² = 25: expansion gives x² − 4x + 4 + y² − 6y + 9 = 25. Why learn the expansion? Some questions supply the circle in general form and some demand the answer in general form, so the two must be interconvertible. Use the identity (a − b)² = a² − 2ab + b². A common mistake is dropping the middle terms, omitting −4x or −6y; another is forgetting the constant terms 4 and 9.
5. Step 5: Simplifying to the general formContinuing: combine the constants on the left of x² − 4x + 4 + y² − 6y + 9 = 25, giving x² + y² − 4x − 6y + 13 = 25; move 25 to the left, giving x² + y² − 4x − 6y − 12 = 0, the general form with D = −4, E = −6 and F = −12. Why must the right side be 0? The general form is defined with all terms moved to one side and zero on the other. A common mistake is a sign error when moving terms, writing +12 instead of −12; substitute one point afterwards to verify.
6. Step 6: Recovering the centre by completing the squareGiven the general form, use completing the square. For x² + y² − 4x − 6y − 12 = 0: move the constant to get x² − 4x + y² − 6y = 12; complete the square in x, x² − 4x = (x − 2)² − 4, and in y, y² − 6y = (y − 3)² − 9; substitute to get (x − 2)² − 4 + (y − 3)² − 9 = 12. Why subtract 4 and 9 back? Whatever is added while completing the square must be subtracted again to keep the equation balanced. A common mistake is forgetting to subtract the added constants, ruining the radius calculation.
7. Step 7: The conclusion of the recoveryTidying the previous step: (x − 2)² + (y − 3)² = 12 + 4 + 9 = 25, the standard form, where the right side 25 is the squared radius, so radius = √25 = 5 and the centre is (2, 3). Check: this matches the equation written directly in Step 2 from the given centre and radius, confirming the algebra. Why practise both directions? Examinations test both, and reading the centre and radius from a general form is a common question. A common mistake is forgetting the square root, answering 25 as the radius; the radius is the positive square root.

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Standard form: (x - h)² + (y - k)² = r²
• Centre (h, k), radius r
• Centre at origin: x² + y² = r²
• When h is negative, minus becomes plus
• Complete the square on general form to find centre and radius