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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:

Finding the midpoint of a line, how to find the middle point of a line. A-Level Maths notes, GCSE Maths. EngineeringNotes.net, EngineeringNotes, Engineering Notes

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)

Angle bisector, Perpendicular bisector, finding diameter of circle, cords in circles. A-Level Maths notes, GCSE Maths. EngineeringNotes.net, EngineeringNotes, Engineering Notes



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:

Completing the square on expanded equation of a circle. How to complete the square on a circle. A-Level Maths. EngineeringNotes.net, EngineeringNotes, Engineering Notes



Intersecting Lines & Circles

A straight line can intersect a circle once, twice, or not at all:

To find out how many intersections there are between a circle and a straight line without sketching it, you can solve simultaneously:

How to work out if a line intersects a circle. Straight line - circle intersections. A-Level Maths Notes. EngineeringNotes, EngineeringNotes.net, Engineering Notes



Circle Theorems

Circle Theorems, Circle maths, Circle theories, circle rules, circle geometry, circle diagrams. GCSE Maths, A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes