Equation of a circle
(x − a)² + (y − b)² = r²: centre (a, b), radius r.
Expanded form: x² + y² + 2gx + 2fy + c = 0. Complete the square to recover centre (−g, −f) and r² = g² + f² − c.
Worked examples
(x−3)² + (y+2)² = 25: centre (3, −2), radius 5.
x² + y² − 4x + 6y + 4 = 0 ⇒ (x−2)² + (y+3)² = 9. Centre (2, −3), r = 3.
Tangent perpendicular to radius at the point of contact.
Frequently asked questions
How to find tangent to circle at point P?
Find centre C; gradient of CP; tangent gradient is the negative reciprocal.
Three points define a circle?
Yes (if not collinear). Set up three equations from the standard form, solve.