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OAN, OMB and APB are straight lines - Edexcel - GCSE Maths - Question 21 - 2017 - Paper 3

Question 21

OAN, OMB and APB are straight lines. AN = 2OA. M is the midpoint of OB. OA = a OB = b AP = kAB where k is a scalar quantity. Given that MPN is a straight line... show full transcript

Worked Solution & Example Answer:OAN, OMB and APB are straight lines - Edexcel - GCSE Maths - Question 21 - 2017 - Paper 3

Step 1

Find vector expression for M

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Answer

Since M is the midpoint of OB, we can express the vector OM in terms of OA and OB:

$OM = \frac{1}{2}(OA + OB) = \frac{1}{2}(a + b)$

Step 2

Express MP using vectors

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Answer

The vector MP can be found as follows:

$MP = O + P - OM = A + P - \frac{1}{2}(A + B) = AP + \left(P - \frac{1}{2}B\right)$

This will provide the components in terms of A and B.

Step 3

Find relationship using straight line condition

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Answer

Since MPN is a straight line, there exists a scalar relationship:

If we express NP in terms of the vectors, we can say:

$NP = k \cdot MP$

This gives the scalar multiple relationship in line.

Step 4

Conclusion for value of k

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Answer

By evaluating the vectors and balancing the components:

We can determine that:

$k = \frac{2}{5}$

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