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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
Set S1 Definition
Answer
The set S1 contains all points P such that the vectors from A to P and from B to P are perpendicular. This means the angle formed by these vectors is 90 degrees, leading to the property of perpendicularity: ar{AP} ullet ar{BP} = 0. This indicates that points P lie on the perpendicular bisector of segment AB.
Step 2
Set S2 Definition
Answer
The set S2 includes all points N such that the distance from A to N is equal to the distance from M to N. This condition can be expressed by the equation: | ar{AN} | = | ar{MN} |. Thus, points N form a circle centered at M with a radius equal to the distance from A to M.
Step 3
Finding the Radius of Circle S
Answer
Since M is the midpoint of AB, we can calculate the distance between A and B: | ar{AB} |, the distance from A to M is therefore given by: | ar{AM} | = rac{| ar{AB} |}{2}. Consequently, the radius of circle S, which is also the distance from M to A, is: r = rac{| ar{AB} |}{2}. Based on this, the answer corresponds to option D: ( rac{ ext{ } ar{AB}}{4} ) as the radius of circle S.