JEE Main 2023 · 31 January · morning · Single correct · +4 / −1

For all z ∈ C on the curve C₁:|z|=4, let the locus of the point z+1/z be the curve C₂. Then :

  1. (a) the curves C₁ and C₂ intersect at 4 points
  2. (b) the curve C₂ lies inside C₁
  3. (c) the curve C₁ lies inside C₂
  4. (d) the curves C₁ and C₂ intersect at 2 points
Answer and solution

Answer (a) the curves C₁ and C₂ intersect at 4 points

Solution

Maths sir · 4 steps

Write z as 4e^(iθ) and you will see z+1/z trace an ellipse that the circle must cut.

  1. Parametrise the point. On |z|=4, put z = 4(cos θ + i sin θ). Then 1/z = 1/4(cos θ - i sin θ), so w = z+1/z = 17/4 cos θ + i 15/4 sin θ.

    C₂: x²/(17/4)² + y²/(15/4)² = 1

  2. Compare the sizes. C₂ is an ellipse with semi-axes 17/4 = 4.25 along the x-axis and 15/4 = 3.75 along the y-axis. The circle C₁ has radius 4, which lies between them. So the ellipse goes outside the circle near the x-axis and stays inside it near the y-axis.
  3. Find the crossings. |w|² = 289/16 cos² θ + 225/16 sin² θ = 225/16 + 4 cos² θ. Setting this equal to 16 gives cos² θ = 31/64, which has four solutions for θ in [0, 2π), one in each quadrant.
  4. So. The circle and the ellipse meet at exactly 4 points, one in each quadrant. The answer is (a).

The idea to keep. For z + 1/z with |z| = r, the locus is an ellipse with semi-axes r + 1/r and r - 1/r. Compare these with r to see how the two curves sit.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Complex Numbers

Asked about 160 times in JEE Main since 2020. This question is from 2023.