Locus in Mathematics

Introduction

Loci: A crucial concept in understanding the distribution of shapes and curves in space, loci describe sets of points that fulfil specific mathematical conditions. They are essential for both theoretical geometry and practical applications, including satellite dish design and calculation of planetary orbits.

Coordinate Geometry Applications:

• Utilised in project designs to ensure precision.
• Improves mapping accuracy in navigation systems.

Definition and Basic Principles

Locus: A set of points meeting defined geometric conditions forms the basis of many geometric principles. These are essential for tackling a range of geometric problems.

Visualising the Concept of Locus

• Circle: Represents a locus of points equidistant from a central point.
• Line: Represents a locus equidistant from two parallel lines.

Key Geometric Shapes and Their Properties

• Circles:
  • Definition: Locus of points equidistant from a central point.
  • Equation: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the centre and $r$ is the radius.
  • Features:
    • Radius: Determines the size of the circle.
    • Diameter: Always double the radius.

• Parabolas:
  • Definition: The locus is equidistant from a fixed point (focus) and a line (directrix).
  • Key Equations: $y^2 = 4ax$ or $x^2 = 4ay$.

• Ellipses & Hyperbolas:
  • Ellipses Definition: Locus with a constant sum of distances from two fixed points (foci).

• Hyperbolas Definition: Locus where the difference in distances to two fixed points (foci) remains constant.

Equation of a Locus from Geometric Conditions

Understanding Loci Equations

Loci equations are commonly encountered in exams. They describe the path of points meeting specified conditions, such as being equidistant from others.

• Identify the geometric condition:
  • Recognise essential characteristics, such as equidistant points.
• Translate to a mathematical expression:
  • Implement the distance formula.
• Simplify and resolve:
  • Equate, simplify, and solve the expression.

Detailed Examples

1. Equidistant from Two Points

• Forms the perpendicular bisector of the line segment joining two points.

Worked Example: Let's find the locus of points equidistant from points A(2,3) and B(6,7).

Step 1: For any point P(x,y) on the locus, PA = PB. Step 2: Using the distance formula: $\sqrt{(x-2)^2 + (y-3)^2} = \sqrt{(x-6)^2 + (y-7)^2}$

Step 3: Square both sides to eliminate the square roots: $(x-2)^2 + (y-3)^2 = (x-6)^2 + (y-7)^2$

Step 4: Expand: $x^2-4x+4+y^2-6y+9 = x^2-12x+36+y^2-14y+49$

Step 5: Simplify: $-4x-6y+13 = -12x-14y+85$ $8x+8y = 72$ $x+y = 9$

The locus is the straight line with equation x + y = 9, which is the perpendicular bisector of the line segment AB.

2. Constant Distance from a Line

• Identifies lines parallel to the given line.

3. Fixed Distance from a Point

• Derives the equation of a circle.

Techniques for Deriving Loci Equations

• Break down each condition using visual aids for clarity.
• Avoid common errors, such as failing to square terms and incorrect algebra.

Principles of Coordinate Geometry for Loci

Distance Formula

• Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Grid and Plot Points

• Steps to Plotting:
  • Identify points and plot them accurately on the grid.
  • Depict loci using the plotted points.

Strategies for Approaching Complex Locus Problems

• Read Carefully: Comprehend each component and constraint thoroughly.
• Visualise: Utilise diagrams to aid understanding and resolve difficulties.
• Step-by-step Analysis: Deconstruct problems, solve systematically, and integrate solutions.