Long Question 1


Question 1:


The above diagram shows triangle OAB. The straight line AP intersects the straight line OQ at R. It is given that OP= 1 4 OB, AQ= 1 4 AB,  OP =4 b ˜  and  OA =8 a ˜ .  

(a) Express in terms of   a ˜  and/ or  b ˜ :
( i ) A P (ii) O Q

(b)(i) Given that A R = h A P , state   A R  in terms of h   a ˜  and  b ˜ .
 (ii) Given that   R Q = k O Q , state  in terms of k,   a ˜  and  b ˜ .

(c) Using   A Q = A R + R Q ,   find the value of h and of k.

Solution
:

(a)(i)
A P = A O + O P A P = O A + O P A P = 8 a ˜ + 4 b ˜


(a)(ii)
O Q = O A + A Q O Q = 8 a ˜ + 1 4 A B O Q = 8 a ˜ + 1 4 ( A O + O B ) O Q = 8 a ˜ + 1 4 ( 8 a ˜ + 4 O P ) O Q = 8 a ˜ + 1 4 ( 8 a ˜ + 4 ( 4 b ˜ ) ) O Q = 8 a ˜ 2 a ˜ + 4 b ˜ O Q = 6 a ˜ + 4 b ˜


(b)(i)
A R = h A P A R = h ( 8 a ˜ + 4 b ˜ ) A R = 8 h a ˜ + 4 h b ˜


(b)(ii)
R Q = k O Q R Q = k ( 6 a ˜ + 4 b ˜ ) R Q = 6 k a ˜ + 4 k b ˜


(c)
A Q = A R + R Q A Q = 8 h a ˜ + 4 h b ˜ + ( 6 k a ˜ + 4 k b ˜ ) A O + O Q = 8 h a ˜ + 4 h b ˜ + 6 k a ˜ + 4 k b ˜ 8 a ˜ + 6 a ˜ + 4 b ˜ = 8 h a ˜ + 6 k a ˜ + 4 h b ˜ + 4 k b ˜ 2 a ˜ + 4 b ˜ = 8 h a ˜ + 6 k a ˜ + 4 h b ˜ + 4 k b ˜ 2 = 8 h + 6 k 1 = 4 h + 3 k ( 1 ) 4 = 4 h + 4 k 1 = h + k k = 1 h ( 2 ) Substitute (2) into (1), 1 = 4 h + 3 ( 1 h ) 1 = 4 h + 3 3 h 4 = 7 h h = 4 7 From (2), k = 1 4 7 = 3 7