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2001

JEE Main 2001 Mathematics question on Complex Numbers and Quadratic Equations

The complex numbers z1,z2z_1, z_2 and z3z_3 satisfying z1−z3z2−z3=1−i32\frac{z_1 - z_3}{z_2 - z_3} = \frac{1 - i\sqrt{3}}{2} are the vertices of a triangle which is
  1. A.
    of area zero
  2. B.
    right angled isosceles
  3. C.
    equilateral
  4. D.
    obtuse angled isosceles
Show answer and solution

Correct option: C

Solution

The ratio (z₁ - z₃)/(z₂ - z₃) = (1 - i√3)/2 has modulus |(1 - i√3)/2| = √(1+3)/2 = 1, so |z₁ - z₃| = |z₂ - z₃|. The argument is arctan(-√3/1) = -60°, meaning z₁ - z₃ is obtained from z₂ - z₃ by a rotation of -60° (and scaling by 1). Since the two sides from z₃ are equal in length and the angle between them is 60°, the triangle is equilateral.

[2001]

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