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NDA 1 2025 Maths, question 52

Topic: Analytical geometry

Q52

If pp and qq are real numbers between 0 and 1 such that the points (p,1)(p,1), (1,q)(1,q) and (0,0)(0,0) form an equilateral triangle, then what is (p+q)(p + q) equal to?

  1. (a) 2\sqrt{2}
  2. (b) 2−1\sqrt{2} - 1
  3. (c) 2−32 - \sqrt{3}
  4. (d) 4−234 - 2\sqrt{3}Correct answer

Solution.

Answer: (d) 4−234 - 2\sqrt{3}

  1. Equality of the two sides from the origin gives p2+1=1+q2p^{2} + 1 = 1 + q^{2}.
  2. Since p,q>0p,q > 0, p=qp = q.
  3. Equating the third squared side to these gives 2(1−p)2=p2+12(1 - p)^{2} = p^{2} + 1, so p2−4p+1=0p^{2} - 4p + 1 = 0.
  4. Only p=2−3p = 2 - \sqrt{3} lies between 0 and 1.
  5. Thus p+q=4−23p + q = 4 - 2\sqrt{3}.