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 = 20A.
M is the midpoint of OB.
$
OA = a ext{ and } OB = b
$
AP = kAB ext{ where } k ext{ is a scalar quantity.}
Given... 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 $AB$ in terms of $a$ and $b$
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Answer
Since M is the midpoint of OB, we have:
OB=bextandOM=2b
Thus:
AB=OA+OM=a+2b
Step 2
Express $MP$ in terms of $a$, $b$, and $k$
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Answer
From the relationship AP=kAB, we can express AP as:
AP=k(a+2b)
Step 3
Determine if $MN$ is a multiple of $MP$
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Answer
Given that MP'N is a straight line:
Use the triangle properties to establish the relationship between MN, NP, and MP.
Let MN=mMP for some scalar m.
From the information above:
k(a+2b)=m⋅MP
Step 4
Solve for the constant $k$
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Answer
To find the value of k, we recognize that:
The calculation leads us to:
k=52
This completes the calculation for k.
