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NTA Abhyas JEE Main2020MathematicsThree Dimensional GeometryPractice

Let the equation of a line is x - 2 1 = y - 3 2 = z - 4 3 . An insect starts flying from P 1 , 3 , 2 in a straight line meeting the given line at a point R a , b , c and then goes to the point Q 6 , 7 , 5 in a straight line such that P R is perpendicular to R Q. Then, the least value of 7 a + b + c is equal to

Options

  1. A105
  2. B45
  3. C10
  4. D7

Correct answer

B. 45

Step-by-step solution

Any general point on the line is λ + 2 , 2 λ + 3 , λ + 4 P R → = < λ + 1 , 2 λ , 3 λ + 2 > R Q → = < 4 - λ , 4 - 2 λ , 1 - 3 λ > From the diagram, P R → ⋅ R Q → = 0 ⇒ λ + 1 4 − λ + 2 λ 4 − 2 λ + 3 λ + 2 1 − 3 λ = 0 ⇒ λ 2 - 3 λ - 4 + 4 λ 2 - 8 λ + 9 λ 2 + 3 λ - 2 = 0 ⇒ 14 λ 2 - 8 λ - 6 = 0 ⇒ λ = 1 , - 3 7 Coordinates of R are 3 , 5 , 7 or 11 7 , 15 7 , 19 7 Least value of 7 a + b + c = 11 + 15 + 19 = 45

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