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Let L be the line passing through the points A(-1, 2, -1) and B(1, 3, 1) . Let N be the foot of the perpendicular drawn from a point P( , 1, ) to the line L . If the distance AN is 3 , the coordinates of N are all positive, the length of the segment PN is 5 , and > , then the value of 2 + 3 is

Options

  1. A7
  2. B8
  3. C5
  4. D6

Correct answer

A. 7

Step-by-step solution

The direction ratios of the line L are (1 - (-1), 3 - 2, 1 - (-1)) = (2, 1, 2) . The equation of the line L passing through A(-1, 2, -1) is x+1 2 = y-2 1 = z+1 2 = r Any point on L can be written as N(2r-1, r+2, 2r-1) . The distance AN is given by AN = (2r-1 - (-1))^2 + (r+2 - 2)^2 + (2r-1 - (-1))^2 = 4r^2 + r^2 + 4r^2 = 3|r| Given AN = 3 , we have 3|r| = 3 |r| = 1 . For r = 1 , N is (1, 3, 1) , which has all positive coordinates. For r = -1 , N is (-3, 1, -3) , which is rejected. The direction ratios of PN are (1-

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