Loci • Topic 1 of 3

Locus of a point at a fixed distance from a point

What is a locus? A locus (plural: loci) is a set of points that satisfies a specific geometric condition or rule. Think of it as a path traced out by a moving point that must obey a strict matching instruction.

What is the locus of a point at a fixed distance from a point? The locus of a point at a fixed distance from a fixed point is a path where every single point on it remains exactly the same distance away from one stationary center point. When a point moves on a flat surface so that its distance from a fixed point stays constant, it traces out a perfect circle.

Let us explore some simple real-world examples:

  • A tied goat: Imagine a goat tied to a wooden stake in the middle of a grassy yard using a tight 3-meter rope. As the goat walks around to eat grass at the very end of the tight rope, the path it travels forms a circle. The wooden stake is the fixed point, and the rope length is the fixed distance.
  • A clock hand: The tip of the minute hand of a wall clock stays a fixed distance away from the central spinning pin. As time passes, the tip moves and traces out a circular locus.
Component NameGeometric MeaningReal-life Equivalent
Fixed PointThe unchanging origin or center of movementThe ground stake / clock pin
Fixed DistanceThe constant length or radiusThe tight rope length / hand length
Traced LocusThe resulting geometric pathThe circular boundary

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DIAGRAM 1: TRACING THE PATH FROM A STATIONARY POINT

                     . - ~ ~ ~ - .   <- Traced Locus (Circle)
                 .                 .
               /                     \
              /       Fixed           \
             ;       Distance          ;
             |       (Radius r)        |
             |O------------------------|P (Moving Point)
             |   (Fixed Center)        |
             ;                         ;
              \                       /
               \                     /
                 .                 .
                     . - _ _ _ - .

DIAGRAM 2: TWO-DIMENSIONAL SPACE RESTRICTION (THE COMPASS)

                Step 1: Fix Center Point (O)
                        * O
                
                Step 2: Move Point P at constant distance 'r'
                       * P3
                    * * P2
                  * O     * ---- r ---- * P1
                    * *
                       * P4
Solved Examples
1
Worked Example
A moving point P maintains a constant distance of 7 cm from a fixed center point O. Describe the exact path traced out by P and calculate its total boundary length. (Take Pi = 22/7).
Solution
  1. Step 1: Identify the geometric shape formed by the rule.*
  2. By definition, a point moving at a constant distance from a single fixed point forms a circle.*
  3. The fixed point O represents the center of the circle, and the fixed distance (7 cm) is its radius.*
  4. Step 2: State the formula for the boundary length (circumference) of this circle.*
  5. Circumference = 2 Pi radius*
  6. Step 3: Substitute the given values into the formula.*
  7. Circumference = 2 (22 / 7) 7*
  8. Circumference = 2 22 1 = 44 cm.*
  9. Answer: The path is a circle with a radius of 7 cm and a boundary length of 44 cm.
2
Worked Example
A dog is tied to a stake fixed in the ground with a rope. The dog runs along its maximum boundary path and completes one full round, covering a total distance of 88 meters. Find the exact length of the tight rope. (Take Pi = 22/7).
Solution
  1. Step 1: Link the physical story to locus properties.*
  2. The maximum boundary path forms a circular locus.*
  3. The total distance of one full round represents the circumference of the circle (88 meters).*
  4. The length of the tight rope represents the unknown radius (r) of the circle.*
  5. Step 2: Set up the algebraic equation using the circumference formula.*
  6. 2 Pi r = 88*
  7. 2 (22 / 7) r = 88*
  8. Step 3: Simplify and isolate the variable r.*
  9. (44 / 7) r = 88
  10. r = 88 (7 / 44)
  11. r = 2 7 = 14 meters.
  12. Answer: The length of the rope is 14 meters.
3
Worked Example
Point A is a fixed point in a plane. A point P moves such that it is always located 5 cm away from A. Another point Q moves such that it is always located 3 cm away from A. Find the minimum and maximum possible straight-line distances between the moving points P and Q at any given moment.
Solution
  1. Step 1: Define the individual loci.*
  2. The locus of P is a circle with center A and radius 5 cm.*
  3. The locus of Q is a circle with the same center A and radius 3 cm.*
  4. These form concentric circles sharing the exact same center.*
  5. Step 2: Find the minimum distance between the two paths.*
  6. The points are closest when they lie on the same straight line extending from center A on the same side.*
  7. Minimum Distance = Radius of P - Radius of Q = 5 cm - 3 cm = 2 cm.*
  8. Step 3: Find the maximum distance between the two paths.*
  9. The points are furthest apart when they lie on opposite sides of the center point A along a single continuous diameter line.*
  10. Maximum Distance = Radius of P + Radius of Q = 5 cm + 3 cm = 8 cm.*
  11. Answer: The minimum distance is 2 cm and the maximum distance is 8 cm.
  12. --

Key Points

  • A locus is a path formed by a collection of all possible points that satisfy a set geometric instruction.
  • The locus of a point at a fixed distance from a stationary point is a circle.
  • The stationary reference point behaves as the center of the resulting circle.
  • The fixed distance constraint dictates the linear radius of the circle.
  • In three dimensions, this same rule would form a hollow sphere instead of a flat circle.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The locus of a point at a fixed distance from a fixed point is a:
Explanation: A circle.
Q2.For that circle, the fixed point is the:
Explanation: The centre.
Q3.The fixed distance is the:
Explanation: The radius.
Q4.A locus is a set of points satisfying a given:
Explanation: A geometric condition.
Practice more MCQs → 📝 Assignment · 35 marks