# STEP 2/2009/8

The non-collinear points A, B, C have position vectors a, b, c respectively. The points P and Q have position vectors p and q, respectively, given by

p = λ a + (1 − λ)b

q = μ a + (1 − μ)c

where 0 < λ < 1. Draw a diagram showing A, B, C, P and Q

Given that CQ.BP = AB.AC, find μ in terms of λ, and show that, for all values of λ, the line PQ passes through the fixed point D, with position vector d given by

d = − a + b + c

# STEP 2/2008/8

The points A and B have position vectors a and b, respectively, relative to the origin O. The points A, B, and O are not collinear. The point P lies on AB between A and B such that

$AP : PB = (1-\lambda):\lambda$

Write down the position vector of P in terms of a, b, and λ

Given that OP bisects the angle AOB, determine λ in terms of a and b, where

a = |a| and b = |b|

The point Q also lies on AB between A and B, and is such that AP=BQ. Prove that

OQ2 − OP2 = (b-a)2