SATHEE: Complex Numbers 4 Question 17

Complex Numbers 4 Question 17

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

$(1983, 2 M)$

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Solution:

Since, $z_{1}, z_{2}$ and origin form an equilateral triangle.

$∵$ if $z_{1}, z_{2}, z_{3}$ from an equilateral triangle, then

$\begin{aligned} z_{1}^{2} + z_{2}^{2} + z_{3}^{2} & = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1} \\ \Rightarrow & z_{1}^{2} + z_{2}^{2} + 0^{2} & = z_{1} z_{2} + z_{2} \cdot 0 + 0 \cdot z_{1} \\ \Rightarrow & z_{1}^{2} + z_{2}^{2} & = z_{1} z_{2} \\ \Rightarrow & z_{1}^{2} + z_{2}^{2} - z_{1} z_{2} & = 0 \end{aligned}$

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Complex Numbers 4 Question 17

17. Prove that the complex numbers z1,z2 and the origin form an equilateral triangle only if z12+z22−z1z2=0.

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