Locus | Lesson Note

01

Activity 1 — Locus at a Fixed Distance from a Point (Circle)

Materials Required

Blank paper (any size) Compass Pencil Ruler Thread

📋 Instructions (Thread Method)

Select a fixed point O somewhere at the middle of the paper.
Cut a piece of thread of any fixed length appropriate for the paper.
Find points $P_1, P_2, P_3, P_4, P_5, \ldots$ around the fixed point O, all at a fixed distance equal to the length of the thread.
Join $P_1, P_2, P_3, \ldots$ smoothly.

📋 Instructions (Compass Method)

Mark a fixed point O at the centre of the paper.
Using a compass set to any fixed radius, cut arcs from O in all directions to mark points $P_1, P_2, P_3, P_4, \ldots$
Continue marking more points all around O at the same radius.
Join all points $P_1, P_2, P_3, \ldots$ to see the resulting path.

Fig. 1 — Points equidistant from fixed centre O form a circle (locus)

Observations

$P_1, P_2, P_3, P_4, \ldots$ are all at a fixed distance from O.
$OP_1 = OP_2 = OP_3 = \cdots$ (all equal)
After joining $P_1, P_2, P_3, \ldots$ we can see a circular shape formed at a fixed distance from O.

✅ Conclusion — Activity 1

Fixed distance from O is the rule.
The circular path formed is actually the locus.

02

Activity 2 — Locus Equidistant from Two Points (Perpendicular Bisector)

Materials Required

Blank paper (any size) Compass Pencil Ruler

📋 Instructions

Draw a line segment AB of any length.
Use a compass; take any measurement but it must be greater than $\dfrac{1}{2}AB$.
Cut arcs from A above and below AB, and intersect arcs from B above and below AB. Mark the intersection points as $P_1$ and $P_2$.
Continue with $> \frac{1}{2}AB$ but with different measurements than the previous step to get $(P_3, P_4)$, $(P_5, P_6)$, $(P_7, P_8)$, and more.
Join $P_1, P_2, P_3, P_4, P_5, \ldots$

Fig. 2 — Points equidistant from A and B lie on the perpendicular bisector of AB (locus)

Observations

$AP_1 = BP_1$, $AP_3 = BP_3$, $AP_5 = BP_5$, $AP_7 = BP_7$
Likewise, $AP_4 = BP_4$, $AP_2 = BP_2$, $AP_6 = BP_6$ etc.
After joining all these points $P_1, P_2, P_3, P_4, P_5, P_6, P_7, \ldots$ we can see a straight line passing exactly through the middle of AB such that $AO = OB$.

✅ Conclusion — Activity 2

The points $P_1, P_2, P_3, P_4, \ldots$ are at equal fixed distances from A and B (use ruler to check).
The rule is to keep $P_1, P_2, P_3, \ldots$ at equal length from A and B.
The straight line (perpendicular bisector of AB) is a path generated by this rule — it is the locus.

03

Summary — Common Loci

#	Rule / Condition	Shape of Locus
1	Point moves at a fixed distance from a fixed point O	Circle with centre O
2	Point moves equidistant from two fixed points A and B	Perpendicular bisector of AB
3	Point moves equidistant from two intersecting lines	Pair of angle bisectors
4	Point moves at a fixed distance from a fixed line	Two parallel lines (one on each side)

04

Definition of Locus

A locus (plural: loci) is a path created by a moving point that follows a fixed rule (condition). Every point on the locus satisfies the given rule, and every point that satisfies the rule lies on the locus.

💡 Simple Way to Remember

Think of a locus as the trail left behind by a point as it moves — like the circular path a pen makes when tied to a fixed nail (that circle is the locus!).

05

Use of Distance Formula in Locus

All we need to do is use the distance formula to find the locus under any given condition (rule).

📐 Two Key Tricks

Keep the moving point as $P(x_2,\, y_2)$ (the one tracing the locus) and the fixed reference point as $A(x_1,\, y_1)$.
No need to take square roots. Use $PA^2$ directly:

Distance Formula (Squared Form)

$$PA^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$$

Using the squared form avoids dealing with square roots and makes simplification much easier.

06

Key Points to Remember

A locus is the set of ALL points satisfying a given condition — not just a few.
To find the equation of a locus, let the moving point be $P(x, y)$ and apply the distance formula (or other condition) between $P$ and the given fixed point(s) or line(s).
Use the squared form $PA^2 = (x_2-x_1)^2 + (y_2-y_1)^2$ to avoid dealing with square roots during simplification.
The resulting equation after simplification is the equation of the locus.
Always verify by checking that a general point on your equation actually satisfies the original rule.

General Distance Formula

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

Use $d^2$ form in locus problems to simplify algebraic work.