JEE Main 2020 · 9 January · morning · Single correct · +4 / −1

Let z be complex number such that
| (z - i)/(z + 2i) | = 1 and |z| = 5/2.
Then the value of |z + 3i| is :

  1. (a) 2√3
  2. (b) √10
  3. (c) 15/4
  4. (d) 7/2
Answer and solution

Answer (d) 7/2

Solution

Maths sir · 4 steps

Modulus 1 means z is equally far from i and -2i, so it sits on the perpendicular bisector.

  1. Read the condition. The ratio has modulus 1 when |z - i| = |z + 2i|. So z is equidistant from the points (0, 1) and (0, -2).
  2. Find the locus. Points equidistant from two points lie on their perpendicular bisector. The midpoint is (0, -1/2) and the segment is vertical, so the bisector is the horizontal line y = -1/2.
  3. Use the modulus. Write z = x - 1/2i. Then |z| = 5/2 gives x² + 1/4 = 25/4, so x² = 6.

    |z + 3i| = √(x² + (-1/2 + 3)²) = √(6 + 25/4) = √(49/4)

  4. So. |z + 3i| = 7/2, which is option (d).

The idea to keep. Whenever you see |(z - a)/(z - b)| = 1, read it as |z - a| = |z - b|: z lies on the perpendicular bisector of a and b. Find that line first, then use the other condition.

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 2020.