Prove that the complex numbers z1, z2...

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1983Moderate

Prove that the complex numbers $z_{1}, z_{2}$ and the origin form an equilateral triangle only if $z_{1}^{2} + z_{2}^{2} - z_{1} z_{2} = 0$ .

Visualized Solution (English)

Visualizing the Triangle

Consider an equilateral triangle in the Argand plane.
Vertices are $z_{1}$ , $z_{2}$ , and the origin $z_{3} = 0$ .
In an equilateral triangle, all sides are equal and internal angles are $6 0^{\circ}$ .

The General Condition

The general condition for three complex numbers $z_{1}, z_{2}, z_{3}$ to form an equilateral triangle is:
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$

Substituting the Origin

Substitute $z_{3} = 0$ into the general condition:
$z_{1}^{2} + z_{2}^{2} + (0)^{2} = z_{1} z_{2} + z_{2} (0) + (0) z_{1}$

Simplifying the Terms

Simplify the equation by evaluating the terms involving zero:
$z_{1}^{2} + z_{2}^{2} + 0 = z_{1} z_{2} + 0 + 0$
$z_{1}^{2} + z_{2}^{2} = z_{1} z_{2}$

Final Rearrangement

Rearrange the terms to match the required form:
Subtract $z_{1} z_{2}$ from both sides:
$z_{1}^{2} + z_{2}^{2} - z_{1} z_{2} = 0$

Key Takeaway

Key Takeaway: For vertices $z_{1}, z_{2}, 0$ to form an equilateral triangle, the condition is $z_{1}^{2} + z_{2}^{2} - z_{1} z_{2} = 0$ .
Next Challenge: Can you derive the condition if the triangle is isosceles with a specific angle at the origin?

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