Circle with Centre (0,0) (Leaving Cert Mathematics): Revision Notes

Circle with Centre (0,0)

What is a Circle?

A circle is the set of all points in a plane that are at a fixed distance, called the radius ($r$), from a fixed point known as the centre.

Equation of a Circle with Centre at $(0, 0)$

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

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

• $(x, y)$: Coordinates of any point on the circle.
• $r$: Radius of the circle.

Key Features of the Circle

1. Radius: Distance from the centre to any point on the circle.
2. Diameter: Twice the radius ($2r$).
3. Circumference: Length of the boundary of the circle, given by: $C = 2\pi r$
4. Area: Space enclosed by the circle, calculated as: $A = \pi r^2$

---

Worked Examples

---

---

Summary

• Definition: A circle is the set of all points equidistant from a fixed centre.
• Standard Equation: $x^2 + y^2 = r^2$
  • $r$ is the radius.
• Key Properties:
  • Radius = $r$, Diameter = $2r$
  • Circumference = $2\pi r$
  • Area = $\pi r^2$
• Use the equation to verify points or find the radius of a circle.