additional mathematics(5r)

7/29/2019 Additional Mathematics(5R)

1/17

Additional MathematicsForm 4

Chapter 6

Coordinate Geometry

By,NurSimah Bachtiar(5R)

Rozana Ramli(5R)


2/17

Coordinate Geometry

Division of a Line SegmentQ divides the line segment PR in the ratio PQ : QR = m : n

nmP(x1, y1) R(x2, y2)Q(x, y)

n

m

R(x2, y2)

P(x1, y1)

Q(x, y)

nm

myny

nm

mxnx 2121 ,Q(x, y) =


3/17

Coordinate Geometry (Ratio Theorem)

nm

myny

nm

mxnx 2121 ,

The point P divides the line segment joining the point M(3,7) and

N(6,2) in the ratio 2 : 1. Find the coordinates of point P.

P(x, y) =

1

2

N(6, 2)

M(3, 7)

P(x, y)

12

)2(2)7(1,

12

)6(2)3(1

3

11,

3

15

115,

3

=

=P(x, y)

=


4/17

Coordinate Geometry

m1.m2 = 1

P

Q

R

S

Perpendicular lines :

Note for

candidates,just sketch

a simple diagram to

help you using therequired formula

correctly.


5/17

Coordinate Geometry

(SPM 2006, P1, Q12)

Diagram 5 shows the straight lineAB which is perpendicular to the straight

line CB at the point B.

The equation ofCB is y = 2x 1 .

Find the coordinates ofB. [3 marks]

mCB = 2

mAB =

Equation of AB is y = x + 4

At B, 2x 1 = x + 4

x = 2, y = 3

So, B is the point (2, 3).

x

y

O

A(0, 4)

C

Diagram 5B

y = 2x 1


6/17

Coordinate Geometry

Given points P(8,0) and Q(0,-6). Find the equation of the

perpendicular bisector of PQ.

344

3

)4(3

4)3( xy

mPQ=

mAB=

Midpoint of PQ = (4, -3)

The equation : 4x + 3y -7 = 0

K1

K1

N1

3

7

3

4 xy

or

P

Q

x

y

O


7/17

TASK: To find the equation of the locus

of the moving point P such that itsdistances from the points A and B are inthe ratio m : n

Sketch a diagram

to help you using

the distance

formula correcty.


8/17

Coordinate Geometry

Find the equation of the locus of the moving point P such that its

distances from the points A(-2,3) and B(4, 8) are in the ratio 1 : 2.(Note : Sketch a diagram to help you using the distance formulacorrectly)

A(-2,3), B(4,8) and m : n = 1 : 2Let P = (x, y)

2

1

B(4, 8)

A(-2, 3)

P(x, y)2 2

2 2 2 2

1

2

2

44 ( 2) ( 3) ( 4) ( 8)

PA

PB

PA PB

PA PBx y x y

3x2 + 3y2 + 24x8y28 = 0


9/17

Find the equation of the locus of the moving point P such that its

distance from the point A(-2,3) is always 5 units. ( SPM 2005)

5

A(-2, 3)

P(x, y)

A(-2,3)Let P = (x, y)

is the equation of locus

of P.

2 24 6 12 0x y x y

2 2 2( 2) ( 3) 5x y


10/17

Coordinate Geometry

Find the equation of the locus of point Pwhich moves such thatit is always equidistant from points A(-2, 3) and B(4, 9).

A(-2, 3)

B(4, 9)

Locus of P

P(x, y)

Constraint / Condition :

PA = PB

PA2

= PB2

(x+2)2 + (y3)2 = (x4)2 + (y9)2

x + y

7 = 0 is the equationof

locus of P.

Note : This locus is actually theperpendicular bisectorof AB


11/17

Solutions to this question by scale drawing will not be accepted.(SPM 2006, P2, Q9)

Diagram 3 shows the triangleAOB where O is the origin.Point Clies on the straight lineAB.

(a) Calculate the area, in units2, of triangleAOB. [2 marks]

(b) Given thatAC : CB = 3 : 2, find the coordinates ofC. [2 marks]

(c) A pointPmoves such that its distance from pointA is always twice its

distance from pointB.

(i) Find the equation of locus ofP,

(ii) Hence, determine whether or not this locus intercepts they-axis.

[6 marks]

x

y

O

A(-3, 4)

Diagram 3C

B(6, -2)


12/17

(SPM 2006, P2, Q9) : ANSWERS

9(a)

= 9

0 6 3 01 10 24 0 0 6 0

0 2 4 02 2

x

y

O

A(-3, 4)

Diagram 3C

B(6, -2)

3

2

9(b) 2( 3) 3(6) 2(4) 3( 2),

3 2 3 2

12 2,5 5

K1

N1

nm

myny

nm

mxnx 2121 ,

Use formula correctly

N1

K1

Use formulaTo find area

Focus

please..


13/17


AP = 2PB

AP2 = 4 PB2

(x+3)2 + (y 4 )2 = 4 [(x 6)2 + (y + 2)2

x2 + y2 18x + 8y + 45 = 0 N1

K1Use distance formula

K1

Use AP = 2PB

x

y

O

A(-3, 4)

C

B(6, -2)

2

1

P(x, y)

AP = 2 2[ ( 3)] ( 4)x y

9(c) (i)


14/17


9(c) (ii)x = 0, y2 + 8y + 45 = 0

b2 4ac = 82 4(1)(45) < 0

So, the locus does not intercept the y-axis.

Use b2 4ac = 0

orAOM

K1

K1 Subst. x = 0 into his locus

N1

(his locus& b2 4ac)


15/17

Given that A(-1,-2) and B(2,1) are fixed points . Point P moves such thatthe ratio of AP to PB is 1 : 2. Find the equation of locus for P.

2 AP = PB

x2 + y2 + 4x + 6y + 5 = 0

K1

J14[ (x+1)2 + (y+2)2 ] = (x -2 )2+ (y -1)2

Coordinate Geometry : the equation of locus

2222)1()2()2()1(2 yxyx

N13x2 + 3y2 + 12x + 18y + 15 = 0

F4


16/17


17/17

additional mathematics(5r)

Documents