Let z₁ and z₂ be two complex numbers such that arg (z₁ - z₂) = π/4 and z₁, z₂ satisfy the equation | z - 3 | = Re(z). Then the imaginary part of z₁ + z₂ is equal to ___________.

Solution

Maths sir · 4 steps

Turn the condition into a parabola; the chord slope then gives you the sum of the y-values directly.

1. Find the locus. Put z = x + iy. Then |z - 3| = Re(z) becomes √((x-3)² + y²) = x. Squaring gives x² - 6x + 9 + y² = x², so y² = 6x - 9.
2. Use both points. Both z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ lie on this parabola. Subtract their equations to get y₁² - y₂² = 6(x₁ - x₂).

   (y₁ - y₂)(y₁ + y₂) = 6(x₁ - x₂)

3. Use the argument. arg(z₁ - z₂) = π/4 means the chord from z₂ to z₁ has slope tan π/4 = 1. So y₁ - y₂ = x₁ - x₂, and neither difference is zero.
4. So. Dividing by y₁ - y₂ gives y₁ + y₂ = 6. The imaginary part of z₁ + z₂ is 6.

The idea to keep. When two points lie on a conic and you know the slope of the chord joining them, subtract the two equations and factor. The difference of squares hands you the sum of the coordinates.

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