Points Inside, Outside or On a Circle (Leaving Cert Mathematics): Revision Notes

Points Inside, Outside or On a Circle

What Determines the Position of a Point Relative to a Circle?

The relative position of a point $P(x, y)$ to a circle is determined using the equation of the circle. For a circle with centre $(h, k)$ and radius $r$, the equation is:

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

The distance $d$ from the centre $(h, k)$ to the point $P(x, y)$ is:

$$d = \sqrt{(x - h)^2 + (y - k)^2}$$

Conditions for a Point's Position

Point Lies on the Circle:

If $d = r$, or equivalently:

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

Point Lies Inside the Circle:

If $d < r$, or equivalently:

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

Point Lies Outside the Circle:

If $d > r$, or equivalently:

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

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Worked Examples

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Summary

• Circle Equation: $(x - h)^2 + (y - k)^2 = r^2$
• Position Determination:
  • On the Circle: $(x - h)^2 + (y - k)^2 = r^2$
  • Inside the Circle: $(x - h)^2 + (y - k)^2 < r^2$
  • Outside the Circle: $(x - h)^2 + (y - k)^2 > r^2$
• Substitute coordinates into the equation to determine the point's position.
• Practice solving these problems to strengthen understanding.