ADDITIONAL MATHEMATICS KSSM F4

1. COORDINATE
GEOMETRY
7.4 EQUATIONS OF
LOCI
Prepared By : Muslimah Binti Mahbob

2. Objective for today’s lesson:
Represent graphically, the locus that satisfies these
conditions:
(i) The distance of a moving point from a fixed
point is constant
(ii) The ratio of a moving point from two fixed
points is constant
And hence, determine the equation of the locus.
Solve problems involving equations of loci
01
02

3. WHAT IS IT
ABOUT LOCUS???
A locus is the set of all points
(usually forming a curve or surface)
satisfying some condition.
For example, the locus of points in
the plane equidistant from a given
point is a circle, and the set of points
in three-space equidistant from a
given point is a sphere.

4. EXPLORATION
TIME

5. LET EXPLORE…
https://www.geogebra.org/m/gmrfv2rs

6. LET EXPLORE…
The distance of a moving point from a fixed point is constant

7. LET EXPLORE…
https://www.geogebra.org/m/ypjbnngv

8. LET EXPLORE…
The ratio of a moving point from two fixed points is constant

9. THE DISTANCE OF A
MOVING POINT
FROM A FIXED POINT
IS CONSTANT

10. EXAMPLE 1

11. EXAMPLE 2
Equation of locus must be
equal to zero & in the
simplest form
   2
1
2
2
1
2 y
y
x
x
D 



   
1
,
2
,
,
,
5 
 Q
y
x
P
PQ
    5
1
2 2
2




 y
x
5
2
1
4
4 2
2





 y
y
x
x
25
2
1
4
4 2
2





 y
y
x
x
0
25
2
1
4
4 2
2






 y
y
x
x
0
20
2
4
2
2




 y
x
y
x
Square both sides

12. THE RATIO OF A
MOVING POINT
FROM TWO FIXED
POINTS IS
CONSTANT

13. EXAMPLE 3
Square both sides
QK
QJ
QK
QJ
QK
QJ
2
3
3
2
3
:
2
:


    2
1
2
2
1
2 y
y
x
x
D 



       2
2
2
2
6
4
2
3
1
3
2
3
y
x
y
x
QK
QJ









       
   
   
0
118
6
50
5
5
0
208
48
32
4
4
90
54
18
9
9
52
12
8
4
10
6
2
9
52
12
8
2
10
6
2
3
12
36
8
16
2
6
9
2
1
3
6
4
2
3
1
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2




















































y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
y
x
x
y
y
x
x
y
x
y
x
Equation of locus must be equal to zero & in the simplest form

14. EXAMPLE 4
Square both sides
RB
AR 2
    2
1
2
2
1
2 y
y
x
x
D 



       2
2
2
2
0
3
2
0
6
2
y
x
y
x
RB
AR









       
   
 
0
12
0
36
3
3
0
36
24
4
4
36
12
9
6
4
36
12
9
6
2
36
12
6
9
2
12
36
0
3
2
0
6
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2












































x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
y
x
x
y
x
x
y
x
y
x
Equation of locus
must be equal to
zero & in simplest
form

15. EXAMPLE 5
Find the equation of locus moving point P such that its
distances from points A(-2 , 0) and B(0 , 4) are the same.
PB
AP 
Square both sides
   2
1
2
2
1
2 y
y
x
x
D 



       2
2
2
2
4
0
0
2 y
x
y
x
PB
AP









       
   
0
3
2
0
12
8
4
0
16
8
4
4
16
8
4
4
16
8
4
4
8
16
4
4
4
0
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2











































y
x
y
x
y
y
x
x
y
x
y
y
x
x
y
x
y
y
x
x
y
x
y
y
x
y
x
x
y
x
y
x
Equation of locus
must be equal to
zero & in simplest
form

16. SOLVING
PROBLEMS

17. EXAMPLE 6

18. EXAMPLE 6

19. EXAMPLE 7
Equation of locus must be
equal to zero & in the simplest
form
   2
1
2
2
1
2 y
y
x
x
D 



    4
4
3 2
2



 y
x
4
16
8
9
6 2
2





 y
y
x
x
16
16
8
9
6 2
2





 y
y
x
x
0
16
16
8
9
6 2
2






 y
y
x
x
0
8
6
2
2



 y
x
y
x
Square both sides

20. EXAMPLE 8
PB
PA
PB
PA
PB
PA
a
2
1
2
1
:
2
:
)
(



Square both sides
   2
1
2
2
1
2 y
y
x
x
D 



       2
2
2
2
0
1
2
0
2
2
y
x
y
x
PB
PA









       
   
 
0
4
0
12
3
3
0
4
8
4
4
4
4
1
2
4
4
4
1
2
2
4
4
2
1
2
4
4
0
1
2
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2












































x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
x
y
x
y
x
x
y
x
x
y
x
y
x
Equation of locus
must be equal to
zero & in the
simplest form

21. EXAMPLE 8
Equation of locus
circle
How can we prove??
 
2
,
2
,
0
4
2
2
C
x
y
x 


By substituting the coordinates C in the equation of the locus
 
 
0
0
0
8
4
4
0
2
4
2
2
2
,
2
,
0
4
2
2
2
2









 C
x
y
x
Since LHS = RHS ( 0 = 0 ), coordinate C (2 , 2) is on the circle.

22. EVALUATION
TIME

23. TRY!!!
DISCUSS THE ANSWER WITH YOUR
TEACHER

24. HOMEWORK

25. SELF PRACTICE 7.10, PG 203, Q1, Q3, Q6 & Q7
(b&c)

26. INTENSIVE PRACTICE 7.4, PG 205, Q2-Q6

27. INTENSIVE PRACTICE 7.4, PG 205, Q2-Q6

28. GOOD LUCK & ALL THE BEST
“Success is the SUM of
SMALL efforts, repeated
DAY IN and DAY OUT.”