The equation (x − a)² + (y − b)² = r², completing the square to find the centre and radius from the general form, and finding the tangent at a point on the circle.
1The standard form
(x − a)² + (y − b)² = r²
centre (a ; b), radius r
centre (a ; b), radius r
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The centre is (a ; b), and the signs flip. (x − 2)² + (y + 3)² = 25 has centre (2 ; −3), not (2 ; 3).
2From the general form: complete the square
Worked ExampleDetermine the centre and radius of x² + y² − 4x + 6y − 12 = 0
- 1Group: (x² − 4x) + (y² + 6y) = 12.Move the constant to the right.
- 2Halve and square each: (−4 ÷ 2)² = 4 and (6 ÷ 2)² = 9.
- 3(x² − 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9.Add to BOTH sides.
- 4(x − 2)² + (y + 3)² = 25.
- 5Centre (2 ; −3), radius 5.r = 25 = 5.
3The tangent at a point
tangent ⊥ radius ⇒ mtangent × mradius = −1
Worked ExampleDetermine the equation of the tangent to the circle above at P(6 ; 0)
- 1Check P is on the circle: (6−2)² + (0+3)² = 16 + 9 = 25. ✓Always verify before you start.
- 2mMP = 0 − (−3)6 − 2 = 34.Gradient of the radius from the centre to P.
- 3mtangent = −43.Negative reciprocal, because the tangent is perpendicular to the radius.
- 4y − 0 = −43(x − 6).Point-gradient form through P.
- 5y = −43x + 8.
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Every circle question that mentions a tangent is really asking about perpendicular gradients. Find the radius gradient first, then flip and change the sign.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Write down the centre and radius of (x − 2)² + (y + 3)² = 25.
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Centre (2 ; −3), radius 5 - 2Determine the centre and radius of x² + y² − 4x + 6y − 12 = 0.
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Completing the square: (x−2)² + (y+3)² = 25 → centre (2 ; −3), r = 5 - 3Show that P(6 ; 0) lies on the circle x² + y² − 4x + 6y − 12 = 0.
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(6−2)² + (0+3)² = 16 + 9 = 25 = r², so P lies on the circle. - 4Determine the gradient of the radius MP where M(2 ; −3) and P(6 ; 0).
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34 - 5Determine the equation of the tangent to the circle at P(6 ; 0).
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m = −43 → y = −43x + 8 - 6Write down the equation of a circle with centre (0 ; 0) and radius 7.
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x² + y² = 49 - 7Determine the radius of x² + y² + 2x − 8y + 8 = 0.
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(x+1)² + (y−4)² = −8 + 1 + 16 = 9 → r = 3 - 8What is the relationship between a tangent and the radius at the point of contact?
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They are perpendicular.
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