Photo AI
Figure 7 shows a sketch of triangle OAB - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 2
Question 12
Figure 7 shows a sketch of triangle OAB. The point C is such that OC = 2OA. The point M is the midpoint of AB. The straight line through C and M cuts OB at the poin... show full transcript
Worked Solution & Example Answer:Figure 7 shows a sketch of triangle OAB - Edexcel - A-Level Maths Pure - Question 12 - 2019 - Paper 2
Step 1
Find CM in terms of a and b
Answer
To find the vector CM, we can use the fact that M is the midpoint of AB. In terms of vectors, we have:
M = rac{OA + OB}{2} = rac{a + b}{2}
Using the position of point C:
Thus, the vector CM can be expressed as:
CM = OC - OM = (2a) - rac{(a + b)}{2}
Simplifying this expression gives:
CM = 2a - rac{1}{2}(a + b) = 2a - rac{1}{2}a - rac{1}{2}b = rac{3}{2}a - rac{1}{2}b
Step 2
Show that ON = ( 2 - rac{3}{2}) a + rac{1}{2} λb
Answer
To find the vector ON, we can use the property that:
We already know that:
From our earlier calculations, let's express CN in terms of the position of M:
Since M is the midpoint, we can express CN as:
Substituting the expression we derived for CM:
CN = λigg( rac{3}{2}a - rac{1}{2}b igg)
Consequently, we have:
ON = 2a + CN = 2a + λ igg( rac{3}{2}a - rac{1}{2}b igg)
Thus:
ON = 2a + rac{3}{2}λa - rac{1}{2}λb
This can be rearranged to show that:
ON = igg(2 - rac{3}{2}λigg) a + igg(- rac{1}{2}λigg) b
Step 3
Hence prove that ON:NB = 2:1
Answer
From the established relationship, we set:
rac{ON}{NB} = rac{ON}{OB - ON}
Substituting from our findings on ON:
= rac{igg(2 - rac{3}{2}λigg) a + igg(- rac{1}{2}λigg) b}{b - igg(2 - rac{3}{2}λigg) a - igg(- rac{1}{2}λigg) b}
The challenge now is simplifying this to show that the ratio indeed equals 2:1. After setting λ = rac{4}{3}, we find that:
ON = rac{4}{3} , NB = 2
So:
, therefore proving the required ratio.