A circle is the set of points at the same distance (the radius) from one point (the centre), and writing that condition in coordinates gives the equation of a circle.
If the centre is (a, b) and the radius is r, every point (x, y) on the circle satisfies (x − a)² + (y − b)² = r², straight from the distance formula.
Even when the equation has been expanded into the form x² + y² + Dx + Ey + F = 0, completing the square in x and in y brings the centre and radius back into view.
In the first picture, changing the centre and radius changes the circle and its equation together; in the second, the way a line meets the circle depends on the distance from the centre to the line.
A circle fixed by its centre and radius
A circle is the set of points at the same distance from the centre. So the centre (a, b) and the radius r are all you need to fix the equation of a circle.
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As the centre (a, b) and radius r change, the circle on the coordinate plane, a point on it and its standard equation change together
💡 From the distance formula to the equation
①The distance between a point (x, y) on the circle and the centre (a, b) is r
②By the distance formula, √((x − a)² + (y − b)²) = r
④Put the coordinates of the green point in the picture into this equation and the two sides always agree
Standard form and general form
Standard form of a circle
(x − a)² + (y − b)² = r²
centre (a, b), radius r
General form of a circle
x² + y² + Dx + Ey + F = 0
Expanding the standard form gives D = −2a, E = −2b, F = a² + b² − r²
📐 From general form to standard form
①Group the x terms to make (x + D/2)²
②Group the y terms to make (y + E/2)²
③Then the centre is (−D/2, −E/2) and r² = (D/2)² + (E/2)² − F
④This must be greater than 0 for a circle; if it is 0 the graph is a single point, and if it is negative there is no graph at all
How a circle and a line can meet
Compare the distance d from the centre to the line with the radius r, and you can tell how many points the circle and line share without solving the simultaneous equations.
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A circle of radius 3 and a line, with the number of intersection points changing as the distance d from the centre to the line changes
Circle and line
d < r: two points, d = r: tangent, d > r: no meeting
d is the distance from the centre of the circle to the line
Equations of tangents
Tangent at a point on the circle
x1·x + y1·y = r²
The tangent to the circle x² + y² = r² at the point (x₁, y₁) on it
Tangent with gradient m
y = mx ± r√(1 + m²)
There are two lines with gradient m that touch the circle x² + y² = r²
Worked examples and unit test
Example 1
Find the equation of the circle with centre (2, −1) and radius 3.
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Substitute a = 2, b = −1, r = 3 into the standard form (x − a)² + (y − b)² = r². Since b is negative, y − (−1) becomes y + 1.
(x − 2)² + (y + 1)² = 3²
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Work out the right-hand side.
(x − 2)² + (y + 1)² = 9
▸ (x − 2)² + (y + 1)² = 9
The coordinates of the centre appear in the equation with their signs reversed. Do not read (y + 1) as a centre with y-coordinate 1.
Example 2
Find the centre and radius of the circle x² + y² − 6x + 4y − 12 = 0.
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Complete the square separately for the x terms and the y terms.
(x − 3)² − 9 + (y + 2)² − 4 − 12 = 0
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Move the constants to the right-hand side to get the standard form.