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NDA 2 2025 Maths, question 63

Topic: Analytical geometry

Q63

What is the value of sin⁡θ\sin \theta if θ\theta is the acute angle between the lines whose equations are px+qy=p+qpx+qy=p+q and p(x−y)+q(x+y)=2qp(x-y)+q(x+y)=2q?

  1. (a) 32\frac{\sqrt{3}}{2}
  2. (b) 34\frac{3}{4}
  3. (c) 12\frac{1}{2}
  4. (d) 12\frac{1}{\sqrt{2}}Correct answer

Solution.

Answer: (d) 12\frac{1}{\sqrt{2}}

  1. Normals to the lines are (p,q) and (p+q,q−p).
  2. The magnitude of their determinant is p²+q²; their lengths are √(p²+q²) and √(2(p²+q²)).
  3. Thus sinθ=1/√2 for nonzero (p,q), giving an acute angle of 45°.