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NDA 1 2024 Maths, question 52

Topic: Analytical geometry

Q52

If (1,−1,2)(1,-1,2) and (2,1,−1)(2,1,-1) are the end points of a diameter of a sphere x2+y2+z2+2ux+2vy+2wz−1=0x^{2}+y^{2}+z^{2}+2ux+2vy+2wz-1=0, then what is u+v+wu+v+w equal to ?

  1. (a) −2-2Correct answer
  2. (b) −1-1
  3. (c) 11
  4. (d) 22

Solution.

Answer: (a) −2-2

  1. The centre is the midpoint of the diameter endpoints, (3/2, 0, 1/2).
  2. Comparing with the general sphere gives u = −3/2, v = 0 and w = −1/2.
  3. Hence u + v + w = −2.
  4. The squared radius is 7/2, consistent with constant term −1.