1.9: In Figure 5, , and is the midpoint of .

4PM1/1R/June/2023 — Question 10 · 11 marks

1.9 diagram 1
In Figure 5, OA→=2a\displaystyle \overrightarrow{OA} = 2\mathbf{a}, OB→=4b\displaystyle \overrightarrow{OB} = 4\mathbf{b} and M\displaystyle M is the midpoint of OA\displaystyle OA.
The point Y\displaystyle Y lies on AB\displaystyle AB such that AY:YB=3:1\displaystyle AY : YB = 3 : 1
The point X\displaystyle X lies on OB\displaystyle OB produced.
(a) Find as simplified expressions in terms of a\displaystyle \mathbf{a} and b\displaystyle \mathbf{b}
(i) AB→\displaystyle \overrightarrow{AB}      (ii) MY→\displaystyle \overrightarrow{MY}
(3)
The points M\displaystyle M, Y\displaystyle Y and X\displaystyle X are collinear.
(b) Find the ratio OB:OX\displaystyle OB : OX
(5)
(c) Find the ratio of (AreaΔYBX):(AreaΔOAX)\displaystyle (\mathrm{Area} \Delta YBX) : (\mathrm{Area} \Delta OAX)
(3)