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Chapter 8: Loci in Two Dimensions

Form 3 Mathematics Bab 8: Loci in Two Dimensions

8.1 Concept of Locus

What is a Locus?

A locus (plural: loci) is a set of points whose locations satisfy one or more specific conditions in two-dimensional space.

8.2 Loci in Two Dimensions

Five Standard Rules for Loci

  • 1. Constant distance from a fixed point:

    The locus of a point moving at a fixed distance $r$ from a fixed point $A$ is a circle with centre $A$ and radius $r$.

  • 2. Equal distance from two fixed points:

    The locus of a point moving at an equal distance from two fixed points $A$ and $B$ is the perpendicular bisector of the line segment $AB$.

  • 3. Constant distance from a straight line:

    The locus of a point moving at a fixed distance $d$ from a straight line $AB$ consists of a pair of parallel lines located distance $d$ on either side of $AB$.

  • 4. Equal distance from two parallel lines:

    The locus of a point moving at an equal distance from two parallel lines $AB$ and $CD$ is a single parallel line passing midway between $AB$ and $CD$.

  • 5. Equal distance from two intersecting lines:

    The locus of a point moving at an equal distance from two intersecting lines $AB$ and $AC$ is the angle bisector of the angle formed by those lines ($\angle BAC$).

8.3 Intersection of Loci

The intersection of two loci is the point or set of points that simultaneously satisfies the conditions of both loci.

  • To find intersection points, construct both individual loci accurately on the same plane.
  • The points where the two loci cross/overlap represent the required position(s).
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