Let A and B be two distinct points in three-dimensional space - HSC - SSCE Mathematics Extension 2 - Question 9 - 2022 - Paper 1
Question 9
Let A and B be two distinct points in three-dimensional space. Let M be the midpoint of AB.
Let S1 be the set of all points P such that $ar{AP} ullet ar{BP} = 0$... show full transcript
Worked Solution & Example Answer:Let A and B be two distinct points in three-dimensional space - HSC - SSCE Mathematics Extension 2 - Question 9 - 2022 - Paper 1
Step 1
Let S1 be the set of all points P such that $ar{AP} \bullet \bar{BP} = 0$
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Answer
The condition ar{AP} \bullet \bar{BP} = 0 implies that the points A, P, and B form a right angle at point P. Thus, S1 represents a plane perpendicular to the line segment AB.
Step 2
Let S2 be the set of all points N such that $| \bar{AN} | = | \bar{MN} |$
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Answer
The point N located such that ∣ANˉ∣=∣MNˉ∣ means that N is equidistant from point A and point M. The locus of such points forms a circle with center at the midpoint M.
Step 3
What is the radius of the circle S?
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Answer
Given that M is the midpoint of segment AB, the distance | ar{AB} | can be expressed as:
| ar{AB} | = | ar{AM} | + | ar{MB} |
Since M is the midpoint, we have:
| ar{MB} | = \frac{| \bar{AB} |}{2}
Thus, the radius of the circle S is:
\frac{| ar{AB} |}{2}
Hence, the answer corresponds to option D: 43∣ABˉ∣.
