Long Questions (Question 9)


Question 9 (6 marks):
Solution by scale drawing is not accepted.
Diagram shows a triangle OCD.
Diagram

(a) Given the area of triangle OCD is 30 units2, find the value of h.

(b)
Point Q (2, 4) lies on the straight line CD.
(i) Find CQ : QD.
(ii) Point P moves such that PD = 2 PQ.
  Find the equation of the locus P.

Solution:
(a)
Given Area of  OCD = 30 1 2  | 0  h 6   0  2   8   0 0 |=30 | ( 0 )( 2 )+( h )( 8 )+( 6 )( 0 )( 0 )( h )( 2 )( 6 )( 8 ) ( 0 )|=60 | 0+8h+00+120|=60 | 8h+ 12|=60 8h+12=60 8h=48 h=6 or  8h+12=60 8h=72 h=9( ignore )


(b)(i)

[ 6( m )+( 6 )( n ) m+n ,  2( m )+( 8 )( n ) m+n ]=( 2, 4 ) 6m6n m+n =2 6m6n=2m+2n 4m=8n m n = 8 4 m n = 2 1 2m+8n m+n =4 2m+8n=4m+4n 2m=4n m n = 4 2 m n = 2 1 Thus, CQ=QD=2:1


(b)(ii)
PD=2PQ ( x6 ) 2 + ( y2 ) 2 =2 ( x2 ) 2 + ( y4 ) 2 ( x6 ) 2 + ( y2 ) 2 =4[ ( x2 ) 2 + ( y4 ) 2 ] x 2 12x+36+ y 2 4y+4=4[ x 2 4x+4+ y 2 8y+16 ] x 2 12x+36+ y 2 4y+4=4 x 2 16x+16+4 y 2 32y+64 The equation of locus P: 3 x 2 +3 y 2 4x28y+40=0