Point P(1,2,3), find the two other points, A & B on the line p=[9,5,1]+t[4,3,1] such that |AP|=|BP|=5.?

Answer 1

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There should be just two such points [picture a sphere with radius 5 centered on P intersected by line p], so we use the distance formula:

√[(9 + 4t - 1)² + (5 + 3t - 2)² + (1 + t - 3)²] = 5

(4t + 8)² + (3t + 3)² + (t - 2)² = 25

16t² + 64t + 64 + 9t² + 18t + 9 + t² - 4t + 4 = 25

26t² + 78t + 77 = 25

26t² + 78t + 52 = 0

t² + 3t + 2 = 0

(t + 1)(t + 2) = 0

t = -1, t = -2

so A = [9,5,1] - [4,3,1] = [5,2,0] and

.... B = [9,5,1] - 2[4,3,1] = [1,-1,-1]

AP = √(16 + 0 + 9) = √25 = 5

BP = √(0 + 9 + 16) = √25 = 5

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Answer 2

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