If z₁ , z₂ are complex numbers such that
Re(z₁) = |z₁ – 1|, Re(z₂) = |z₂ – 1| , and
arg(z₁ - z₂) = π/6, then Im(z₁ + z₂ ) is equal to :

Solution

Maths sir · 3 steps

Both points lie on one parabola, so the slope of the chord joining them gives y₁ + y₂ directly.

1. Find the locus. Put z = x + iy. The condition x = |z - 1| becomes x = √((x-1)² + y²). Squaring gives x² = x² - 2x + 1 + y², so y² = 2x - 1. Both z₁ and z₂ lie on this parabola.
2. Use the argument. arg(z₁ - z₂) = π/6 means the chord from z₂ to z₁ has slope tan π/6 = 1/√3. From the parabola, 2x₁ - 2x₂ = y₁² - y₂².

   (y₁ - y₂)/(x₁ - x₂) = 2(y₁ - y₂)/(y₁² - y₂²) = 2/(y₁ + y₂) = 1/√3

3. So. y₁ + y₂ = 2√3, and y₁ + y₂ is exactly Im(z₁ + z₂). The answer is (d), 2√3.

The idea to keep. For two points on y² = 4ax, the chord slope is 4a/(y₁ + y₂). Whenever a question gives the direction of a chord, it is really telling you the sum of the y-coordinates.

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