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Pure · Coordinate geometry

Chapter 1 · 3

The idea

The equation of a circle

Why (x − a)² + (y − b)² = r² is just the distance formula squared, reading the centre and radius off it, and completing the square to recover them from the expanded form x² + y² + 2gx + 2fy + c = 0.

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Pure · Coordinate geometry

The equation of a circle

Why (x − a)² + (y − b)² = r² is just the distance formula squared, reading the centre and radius off it, and completing the square to recover them from the expanded form x² + y² + 2gx + 2fy + c = 0.

Why it works

The distance formula, squared

A circle is the set of all points a fixed distance rr (the radius) from a fixed point (a,b)(a, b) (the centre). Take any point (x,y)(x, y) on it: the distance from the centre is (x−a)2+(y−b)2\sqrt{(x - a)^2 + (y - b)^2}, and that must equal rr.

Square both sides and you have the equation of the circle:

(x−a)2+(y−b)2=r2(x - a)^2 + (y - b)^2 = r^2

So it is literally the length formula with the square root undone.

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