Circles

Interactive Learning Module

Deriving the Standard Form

Consider a circle with centre C(a, b) and a point P(x, y) on its circumference.

Expanding to General Form

Let's expand the standard form step by step.

Converting Back: Completing the Square

Let's work through an example: \(x^2 + y^2 - 6x + 4y - 3 = 0\)

Step 1 of 3
Standard Form:
General Form:
Centre: (0, 0) | Radius: 3

Guiding Questions:

  • What happens to the circle when you increase r?
  • How does changing a affect the position of the circle?
  • Can you position the circle so it passes through the origin? What equation does this give?