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Let z₁ and z₂ be nth roots of unity which subtend a right angle at the origin, then n must be of the form :
- a)4k+1
- b)4k+2
- c)4k+3
- d)4k
Correct answer is option 'D'. Can you explain this answer?
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Explanation:
To understand why the answer is option 'D', let's break down the problem step by step.
1. Roots of Unity:
Roots of unity are the complex numbers that satisfy the equation z^n = 1, where n is a positive integer. These roots are equally spaced on the unit circle in the complex plane.
2. Subtending a Right Angle:
When two complex numbers z and z' subtend a right angle at the origin, the vectors representing these complex numbers form a right angle.
3. Representation in the Complex Plane:
Let's represent z and z' in the complex plane as z = cos(θ) + i sin(θ) and z' = cos(φ) + i sin(φ), where θ and φ are the arguments of z and z' respectively.
4. Geometric Interpretation:
For the vectors representing z and z' to form a right angle, the difference between their arguments should be π/2 radians (90 degrees). Therefore, we have φ - θ = π/2.
5. Expressing z and z' in Terms of n:
Since z and z' are nth roots of unity, their arguments can be expressed as θ = 2πk/n and φ = 2πm/n, where k and m are integers.
6. Substituting in the Equation:
Substituting the values of θ and φ in the equation φ - θ = π/2, we have (2πm/n) - (2πk/n) = π/2.
Simplifying, we get 2π(m - k)/n = π/2.
7. Simplifying the Equation:
Canceling out π and rearranging the equation, we have (m - k)/n = 1/4.
8. Conclusion:
From the above equation, we can conclude that n must be a multiple of 4 (4k), since (m - k) can be any integer and 1/4 is a fixed value.
Therefore, the correct answer is option 'D' - n must be of the form 4k, where k is an integer.
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