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In a triangle PQR , a point X divides the side PQ internally in the ratio 2:1 and a point Y divides the side QR internally in the ratio 3:1 . If the line segment XY is extended to intersect the line PR at a point Z , then the value of k , where the length PZ = k ZR , is equal to

Options

  1. A6
  2. B2
  3. C3
  4. D1

Correct answer

A. 6

Step-by-step solution

Let the position vectors of the vertices P, Q, and R be p , q , and r respectively. The point X divides PQ internally in the ratio 2:1 , so its position vector is: x = p + 2 q 3 The point Y divides QR internally in the ratio 3:1 , so its position vector is: y = q + 3 r 4 Since Z lies on the line passing through X and Y , its position vector z can be written as an affine combination of x and y : z = x + (1 - ) y z = ( p + 2 q 3 ) + (1 - ) ( q + 3 r 4 ) Grouping the terms by p , q , and r : z = 3 p + ( 2 3 + 1 - 4 )

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