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• A-Level Maths

# Circles

Often when trying to find the equation of a circle, you will be given a line that intersects with the circle twice (it may be the diameter). To work out the circle from this you need to know the midpoint: The perpendicular bisector of a cord will always pass through the centre of a circle. To find this, find the midpoint of the cord and find the equation of the line perpendicular to it (the negative reciprocal of the gradient) ## Equation of a Circle

The points on a circle are all related through the equation of a circle.

For a circle with centre (0,0) and radius r, the equation is:

x² + y² = r²

For a circle with centre (a, b) and radius r, the equation is:

(x-a)² + (y-b)² = r²

### Expanded Form

Sometimes, the equation is given or needed in expanded form:

x² + y² + 2fx + 2gy + c = 0
• (-f, -g) is the centre of the circle

• (f² + g² -c) is the radius

To go back from expanded form to the standard factorised form for an equation of a circle, it is quickest to complete the square: 