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

Circle with Centre other than (0,0)

What is the Equation of a Circle?

A circle is defined as the set of all points that are equidistant from a fixed point, called the centre. If the centre of the circle is at $(h, k)$ and the radius is $r$, the equation of the circle is:

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

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

Key Components of the Equation

1. Centre: The values $h$ and $k$ define the location of the circle's centre.
2. Radius: The fixed distance $r$ between the centre and any point on the circle.
3. Expanded Form: The equation can be expanded to:

$$x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 = 0$$

Applications

• Verify whether a point lies on the circle by substituting its coordinates into the equation.
• Solve problems involving intersections of a line and a circle.

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Worked Examples

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Summary

• Standard Equation: For a circle with centre $(h, k)$ and radius $r$:

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

• Components:
  • Centre: $(h, k)$
  • Radius: $r$
• Applications: Verify points, solve line-circle intersections.
• Practice using the equation to solidify understanding.