Question
Let $Z_1$ and $Z_2$ be any two complex numbers such that $Z_1^2 + Z_2^2 + Z_1Z_2 = 0$.
What is the value of \(\left|\frac{Z_1}{Z_2}\right|\)?
This question was previously asked in
NDA 2 2024 GAT Question Paper (01-Sep-2024)
The correct answer is
1
To solve the problem, we need to find the value of \(\left|\frac{Z_1}{Z_2}\right|\)given the condition \(Z_1^2 + Z_2^2 + Z_1Z_2 = 0\).
- Let's assume the complex numbers \(Z_1\)and \(Z_2\)as \(Z_1 = a + bi\)and \(Z_2 = c + di\), where \(a, b, c,\)and \(d\)are real numbers.
- The given condition is \(Z_1^2 + Z_2^2 + Z_1Z_2 = 0\).
- One way to solve this equation is to treat it algebraically and find a relationship between the magnitudes of the complex numbers:
- Rewrite the equation in terms of the magnitudes: \(|Z_1|^2 + |Z_2|^2 + 2|Z_1||Z_2|\cos(\theta) = 0\), where \(\theta\)is the angle between \(Z_1\)and \(Z_2\).
- Since all terms are squared or involve cosine, we simplify assuming angles make \(\cos(\theta) = -1\)(i.e., they are collinear with opposite signs):
- Then, \(|Z_1|^2 + |Z_2|^2 - 2|Z_1||Z_2| = 0\)simplifies to \(( |Z_1| - |Z_2|)^2 = 0\).
- Thus, \(|Z_1| = |Z_2|\).
- Therefore, the value of \(\left|\frac{Z_1}{Z_2}\right|\)is 1.
Thus, the correct option is \(\textbf{1}\).
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