MathsICSE Class 10 · Loci

Locus of a Point at a Fixed Distance from a Fixed Point: the Circle (ICSE Class 10)

Q: What is the locus of a point at a fixed distance from a fixed point?

A circle whose centre is the fixed point and whose radius is the fixed distance. This is the first locus theorem in the ICSE syllabus.

Q: How many points on a line are at a given distance from a point?

Two if the line is closer to the point than the given distance, one if it is exactly that distance away (the line touches the circle), and none if it is further.

Q: What is the locus of points at a fixed distance from a line?

A pair of lines parallel to the given line, one on each side, at that distance.

Updated 15 October 2026 · Fact-checked

The locus of a point at a fixed distance r from a fixed point O is the circle with centre O and radius r. To find points satisfying another condition as well, draw both loci and mark where they meet. A line closer to O than r cuts it twice.

Every point exactly r from O lies on the circle; a line at distance d from O meets it twice if d < r, once if d = r, never if d > r.

Illustration drawn for QuestionBuddy.

Understand Locus at a Fixed Distance from a Point

This is the first locus theorem in the CISCE syllabus: the locus of a point at a fixed distance from a fixed point is a circle with the fixed point as centre and the fixed distance as radius.

Exactly, not within. 'At a distance of 3 cm from $O$ ' is the circle. 'Within 3 cm of $O$ ' is the whole region inside and on the circle (a disc), which is a region, not a curve.

Construction. Set the compass to the radius, put the point on $O$ and draw the complete circle unless the question limits it. In 2024 Q10(iii)(c) the examiners reported candidates who did not identify the locus at a fixed distance $O A$ from $O$ as a circle.

Circle and line. Let $d$ be the perpendicular distance of a line from $O$ and $r$ the radius.

$d < r$ : the line cuts the circle at two points, each $r^{2} - d^{2}$ from the foot of the perpendicular.
$d = r$ : one point (the line touches the circle).
$d > r$ : no point.

Circle and circle. Two circles with centres $A$ , $B$ and radii $r$ , $s$ meet in two points when $∣ r - s ∣ < A B < r + s$ , one point when $A B = r + s$ (or $∣ r - s ∣$ ), and none otherwise.

Related locus: fixed distance from a line. Points at a fixed distance $d$ from a line $l$ form **two lines parallel to $l$ **, one on each side, at distance $d$ . Combined with a circle, count the meeting points line by line.

Fixed distance from a segment or region. The locus of the centre of a coin of radius $r$ rolling round a fixed circle of radius $R$ is a circle of radius $R + r$ ; the centre of a wheel rolling along a straight road moves along a line parallel to the road at height $r$ .

Where this appears in exams. It is usually one step of a longer locus question: construct a circle, then find where it meets a perpendicular bisector or an angle bisector.

Locus at a Fixed Distance from a Point: formulas to remember

Locus theorem: $O P = r ⟺ P lies on the circle with centre O, radius r$; Proof not required.
Circle and line: $d < r : 2, d = r : 1, d > r : 0 points$; $d$ = distance of the line from the centre.
Half-chord: $r^{2} - d^{2}$; Distance of each point from the foot of the perpendicular.
Two circles meet: $∣ r - s ∣ < A B < r + s \Rightarrow 2 points$; Centres $A$ , $B$ ; radii $r$ , $s$ .
Fixed distance d from a line: $two lines parallel to l at distance d$; One on each side.

How to solve Locus at a Fixed Distance from a Point questions

Turn each condition into a locus, draw them, and use distances to count meetings before constructing.

1For 'at distance $r$ from $O$ ', draw the circle centre $O$ , radius $r$ .
2For any second condition, draw its locus (perpendicular bisector, angle bisector, parallel lines, another circle).
3Predict the number of meeting points using $d$ and $r$ (or the sum and difference of radii).
4Mark and name the points; measure what is asked.
5Check measurements with Pythagoras ( $r^{2} - d^{2}$ ) where possible.

Locus at a Fixed Distance from a Point in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

A circle as the boundary of a reach

Suppose a garden sprinkler fixed at one point throws water exactly 3 m in every direction.

Describe the edge of the wetted patch and find the area it waters (take $π = 3.14$ ).

Try it first, then show the solutionSolution

The edge is the locus of points 3 m from the sprinkler: a circle of radius 3 m.
Area $= 3.14 \times 3^{2} = 28.26$ m².

Answer: A circle of radius 3 m; 28.26 m² watered.

Memory hook: Fixed distance from a point = a circle.

Counting points in range

Imagine a radio mast whose signal reaches exactly 50 km, and two villages 30 km and 60 km from it.

Which village is inside the range, and what shape is the boundary of the range?

Try it first, then show the solutionSolution

The boundary is the circle of radius 50 km about the mast.
30 km < 50 km: inside. 60 km > 50 km: outside.

Answer: The village 30 km away is in range; the boundary is a circle of radius 50 km.

Memory hook: Compare the distance with the radius.

Tricks to remember Locus at a Fixed Distance from a Point

Short memory aids. Say each one aloud once, then use it on a question the same day.

Compass = locus
Whenever a condition says 'at a distance … from a point', reach for the compass and draw a full circle.
Use it when: Recognising the locus.
d versus r
Closer than the radius: two hits; equal: one touch; further: miss.
Use it when: Counting points on a line.
Distance from a line: railway tracks
Points 2 cm from a line make two parallel 'rails', one each side.
Use it when: The related locus from a line.

Locus at a Fixed Distance from a Point: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Naming the locus · 1 mark · MCQBoard level

The locus of a point that is always 4 cm from a fixed point $O$ is:
(a) a line 4 cm from $O$
(b) a circle with centre $O$ and radius 4 cm
(c) a circle with diameter 4 cm through $O$
(d) a pair of parallel lines

Try it first, then show the solutionSolution

Fixed distance from a fixed point: circle with that point as centre and that distance as radius.

Answer: (b)

Example 2: Points on a line at a given distance · 3 marks · Short answerBoard level

A point $O$ is 6 cm from a straight line $l$ . How many points on $l$ are 10 cm from $O$ ? Find their distances from the foot of the perpendicular from $O$ , and from each other.

Try it first, then show the solutionSolution

The points lie on the circle centre $O$ , radius 10 cm; $d = 6 < 10$ , so there are two.
Each is $1 0^{2} - 6^{2} = 8$ cm from the foot.
They are $16$ cm apart.

Answer: Two points; 8 cm from the foot; 16 cm apart.

Example 3: Two circles · 3 marks · Long answerHarder board level

Draw $A B = 7$ cm. Construct the locus of points 4 cm from $A$ and the locus of points 5 cm from $B$ .
(a) How many points satisfy both conditions? Why?
(b) Measure the distance between them.

Try it first, then show the solutionSolution

Draw the circles: centre $A$ radius 4 cm, centre $B$ radius 5 cm.
(a) $5 - 4 = 1 < 7 < 4 + 5 = 9$ , so the circles cross at two points, one on each side of $A B$ .
(b) The two points are about 5.6 cm apart (each is about 2.8 cm from $A B$ ).

Answer: (a) Two (b) about 5.6 cm

Example 4: Region reachable by a tethered dog · 3 marks · Short answerHarder board level

A dog is tied by a 3.5 m rope to a ring at the middle of a long straight wall. Describe the boundary of the region it can reach, and find its area (take $π = \frac{22}{7}$ ).

Try it first, then show the solutionSolution

Points 3.5 m from the ring form a circle, but the wall blocks one side.
So the boundary is a semicircle of radius 3.5 m with the wall as its diameter.
Area $= \frac{1}{2} \times \frac{22}{7} \times 3. 5^{2} = 19.25$ m².

Answer: A semicircle of radius 3.5 m; 19.25 m²

Example 5: Circle through three points from loci · 4 marks · Short answerHOTS / competency

In $△ A B C$ , $∠ A B C = 90°$ , $A B = 5$ cm and $B C = 12$ cm. $O$ is equidistant from $A$ , $B$ and $C$ . Describe the locus of points at a distance $O A$ from $O$ , and find its radius. Explain why it passes through $B$ and $C$ .

Try it first, then show the solutionSolution

$O$ is the circumcentre, the mid-point of the hypotenuse $A C$ .
$A C = 25 + 144 = 13$ cm, so $O A = 6.5$ cm.
The locus is the circle centre $O$ , radius 6.5 cm. Since $O B = O C = O A$ , the circle passes through $B$ and $C$ too.

Answer: The circle centre $O$ , radius 6.5 cm (the circumcircle).

Example 6: Case study: a ship and a lighthouse · 4 marks · Case-basedHOTS / competency

A lighthouse beam can be seen up to 10 km away. A ship sails in a straight line that passes 6 km from the lighthouse at its closest.
(a) Describe the boundary of the region where the beam is visible.
(b) For what length of its course can the ship see the beam?
(c) At 20 km/h, for how many minutes is the beam visible?

Try it first, then show the solutionSolution

(a) A circle of radius 10 km centred on the lighthouse.
(b) The course is a line at distance 6 km from the centre: it cuts the circle in a chord of length $2 1 0^{2} - 6^{2} = 16$ km.
(c) Time $= \frac{16}{20}$ h $= 48$ minutes.

Answer: (a) A circle of radius 10 km (b) 16 km (c) 48 minutes

Where students lose marks in Locus at a Fixed Distance from a Point

The slips examiners see most often, and what to write instead.

Mistake: Not recognising the circle
A fixed distance from a fixed point is always a circle.
Wrong working
Part (c) of 2024 Q10(iii) drawn as a line through $O$ .
Right working
Circle with centre $O$ and radius $O A$ .
Do this: Translate each condition into words: 'distance from a point → circle'.
Usually costs: Reported in 2024 Q10(iii).
Mistake: Drawing only part of the circle
The locus is the whole circle unless limited.
Wrong working
A small arc near one point.
Right working
A complete circle, so all intersections appear.
Do this: Complete every locus circle.
Usually costs: Missed intersection points.
Mistake: Only one parallel line for distance from a line
Points at distance $d$ lie on both sides.
Wrong working
One line 4 cm above $A B$ .
Right working
Two lines, 4 cm above and 4 cm below $A B$ .
Do this: Always ask 'which side?'
Usually costs: Half the answer points.

Exam tips

2024 Q10(iii)(c) asked for the locus of points at a fixed distance $O A$ from $O$ : the circle with centre $O$ through $A$ .
Before constructing, compare distances to predict how many points you should find.
Label each locus and each intersection point clearly.
Use a sharp pencil and a firm compass so circles close neatly.

Practice questions on Locus at a Fixed Distance from a Point

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The locus of the centre of a wheel of radius 30 cm rolling along a straight level road is:
(a) a circle of radius 30 cm
(b) a line parallel to the road, 30 cm above it
(c) the road itself
(d) a line perpendicular to the road
Hint
The centre stays 30 cm from the road.
Answer
(b)
Q2 · 1 mark · Short answerBoard level
A line is 3 cm from a point $O$ . How many points on the line are 3 cm from $O$ ?
Hint
$d = r$ .
Answer
One: the line touches the circle.
Q3 · 3 marks · Short answerHarder board level
$A$ and $B$ are 10 cm apart. How many points are 6 cm from both $A$ and $B$ ? How far is each from $A B$ ?
Hint
They lie on the perpendicular bisector, 5 cm from $A$ along $A B$ .
Answer
Two; each $36 - 25 = 11 \approx 3.3$ cm from $A B$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The locus of points within 2 cm of a fixed point $O$ is a circle of radius 2 cm.
Reason (R): The locus of points exactly 2 cm from $O$ is a circle of radius 2 cm.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
'Within' includes the inside.
Answer
(b). 'Within 2 cm' is the whole disc, not just its boundary.
Q5 · 3 marks · Short answerHarder board level
$A B = 6$ cm. The locus of points 4 cm from $A$ meets the locus of points equidistant from $A$ and $B$ at $P$ and $Q$ . Find $P Q$ .
Hint
$P$ is 3 cm along $A B$ from $A$ and $16 - 9$ from $A B$ .
Answer
$P Q = 27 \approx 5.3$ cm
Q6 · 3 marks · Short answerHOTS / competency
A point $O$ is 4 cm from a line $l$ . How many points are 2 cm from $l$ and 5 cm from $O$ ?
Hint
The two parallel lines are 2 cm and 6 cm from $O$ .
Answer
The near line (2 cm from $O$ ) cuts the circle twice; the far line (6 cm from $O$ ) misses it. Two points.
Q7 · 3 marks · Short answerHarder board level
$A$ and $B$ are 4 cm apart. Find the number of circles of radius 3 cm that pass through both $A$ and $B$ , and the distance of each centre from $A B$ .
Hint
Centres are 3 cm from $A$ and 3 cm from $B$ .
Answer
Two circles; each centre is $9 - 4 = 5 \approx 2.2$ cm from $A B$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Locus of points 5 cm from $O$ ?Show answer
The circle with centre $O$ and radius 5 cm.
2.Line 7 cm from $O$ ; circle radius 5 cm. Points of intersection?Show answer
None.
3.Locus of points 2 cm from a line?Show answer
Two lines parallel to it, 2 cm away on each side.
4.Circles radii 3 and 4, centres 5 apart: how many meeting points?Show answer
Two (since $1 < 5 < 7$ ).
5.Half-chord when $r = 13$ and $d = 5$ ?Show answer
12

Locus at a Fixed Distance from a Point: frequently asked questions

What is the locus of a point at a fixed distance from a fixed point?

A circle whose centre is the fixed point and whose radius is the fixed distance. This is the first locus theorem in the ICSE syllabus.

How many points on a line are at a given distance from a point?

Two if the line is closer to the point than the given distance, one if it is exactly that distance away (the line touches the circle), and none if it is further.

What is the locus of points at a fixed distance from a line?

A pair of lines parallel to the given line, one on each side, at that distance.