The Basics (Leaving Cert Mathematics): Revision Notes

Equation of a Circle

What is the Equation of a Circle?

A circle is the set of all points that are at a fixed distance (radius $r$) from a fixed point called the centre. In coordinate geometry, the equation of a circle can be represented based on the position of its centre.

Standard Form of the Equation of a Circle

Centre at $(0, 0)$:

If the centre is the origin $(0, 0)$ and the radius is $r$, the equation is:

$$x^2 + y^2 = r^2$$

Centre at $(h, k)$:

For a circle with centre $(h, k)$ and radius $r$, the equation is:

$$(x - h)^2 + (y - k)^2 = r^2$$

Expanded Form of the Equation

The expanded form of a circle's equation, derived from the standard form, is:

$$x^2 + y^2 + 2gx + 2fy + c = 0$$

Here:

• $−g = h$ and $-f = k$, so the centre is $(-g, -f)$
• The radius is calculated as $r = \sqrt{g^2 + f^2 - c}$

Applications

• Determining if a point lies on the circle by substituting its coordinates into the equation.
• Solving problems involving tangents, chords, or intersections with lines.

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Worked Examples

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Summary

• Equations of a Circle:
  • Centre $(0, 0): x^2 + y^2 = r^2$
  • Centre $(h, k): (x - h)^2 + (y - k)^2 = r^2$
  • Expanded Form: $x^2 + y^2 + 2gx + 2fy + c = 0$
• Key Calculations:
  • Centre from $-g, -f$
  • Radius from $r = \sqrt{g^2 + f^2 - c}$
• Practice identifying centres and radii from various forms of the equation.