Circles

The equation of a circle is a core topic in the Edexcel 9MA0 specification. You need to understand both forms of the equation, find centres and radii, determine whether points lie inside, on or outside a circle, and work with tangents to circles.

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The Equation of a Circle

Standard Form

A circle with centre (a, b) and radius r has the equation:

(x − a)² + (y − b)² = r²

Example: The circle with centre (3, −2) and radius 5 has equation (x − 3)² + (y + 2)² = 25.

Expanded (General) Form

Expanding the standard form gives:

x² + y² + 2gx + 2fy + c = 0

where the centre is (−g, −f) and the radius is √(g² + f² − c), provided g² + f² − c > 0.

Example: Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.

Compare: 2g = −6, so g = −3; 2f = 4, so f = 2; c = −12.

Centre = (−g, −f) = (3, −2). Radius = √(9 + 4 − (−12)) = √25 = 5.

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