Example (1): Writing the equation from the centre and radius

• Find the equation of the circle with centre (2, −3) and radius 4, then check whether the point (2, 1) lies on it.
• Solution:

1. (x − 2)² + (y + 3)² = 4² = 16.
2. (2 − 2)² + (1 + 3)² = 0 + 16 = 16.
3. Both sides are equal, so the point (2, 1) lies on the circle.

Example (2): Converting the general equation to standard form

• Find the centre and the radius of the circle whose equation is x² + y² − 4x + 6y − 3 = 0.
• Solution:

1. 2g = −4, so g = −2; 2f = 6, so f = 3; and c = −3.
2. Centre = (−g, −f) = (2, −3).
3. r² = g² + f² − c = 4 + 9 + 3 = 16, so r = 4.
4. (x − 2)² + (y + 3)² = 16.

Example (3): A circle with a known diameter

• Find the equation of the circle whose diameter has end points A(1, 2) and B(5, 6).
• Solution:

1. ((1 + 5) ÷ 2, (2 + 6) ÷ 2) = (3, 4).
2. AB = √((5 − 1)² + (6 − 2)²) = √32 = 4√2, so the radius is 2√2.
3. r² = 8, so the equation is (x − 3)² + (y − 4)² = 8.
4. (1 − 3)² + (2 − 4)² = 4 + 4 = 8.

Example (4): A circle centred at the origin, and the position of a point

• Find the equation of the circle with centre at the origin that passes through (3, 4), then determine the position of the point (4, 4) relative to it.
• Solution:

1. r² = 3² + 4² = 25, so the equation is x² + y² = 25.
2. 4² + 4² = 32.
3. Since 32 > 25, the point (4, 4) lies outside the circle.

Example (5): A circle touching the y-axis

• Find the equation of the circle with centre (−1, 2) that touches the y-axis.
• Solution:

1. The distance from the centre to the y-axis is |−1| = 1. Since the circle touches the axis, this distance equals the radius.
2. The equation is (x + 1)² + (y − 2)² = 1.

Example (6): An equation that does not represent a real circle

• Decide whether x² + y² + 2x − 4y + 6 = 0 represents a real circle.
• Solution:

1. g = 1, f = −2, c = 6.
2. g² + f² − c = 1 + 4 − 6 = −1 < 0.
3. So the equation does not represent a real circle, because no point satisfies it.