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 $ar{AP} ullet ar{BP} = 0$. Let S2 be the set of all points N such that $|ar{AN}| = |ar{MN}|$. The intersection of S1 and S2 is the circle S. What is the radius of the circle S?

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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

Let S1 be the set of all points P such that AP ⋅ BP = 0

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Answer

This implies that the points P lie on the perpendicular bisector of the segment AB.

Step 2

Let S2 be the set of all points N such that |AN| = |MN|

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Answer

Since M is the midpoint of AB, the points N are at equal distance from A and M.

Step 3

Determine the intersection of S1 and S2 which is the circle S

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Answer

The intersection will be a circle with its center at M and radius equal to half the distance of AB because points on S2 are equidistant from A and M.

Step 4

What is the radius of the circle S?

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Answer

The radius of the circle S is given by $\frac{\sqrt{3} |AB|}{4}$ , which corresponds to option D.

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