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 \(\textbf{1}\).
Similar Questions
- If \(z\) is any complex number and \(iz^3 + z^2 - z + i = 0\), where \(i = \sqrt{-1}\), then what is the value of \((|z|+1)^2\)?
- Let \(z_1\) and \(z_2\) be two complex numbers such that \(\left|\frac{z_1+z_2}{z_1-z_2}\right| = 1\), then what is \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\) equal to ?
- If \(\omega \ne 1\) is a cube root of unity, then what are the solutions of \((z-100)^3 + 1000 = 0\) ?
- What is \((1+i)^4 + (1-i)^4\) equal to, where \(i = \sqrt{-1}\)?
- If \(\omega \neq 1\) is a cube root of unity, then what is \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) equal to?
- What is the value of the sum
\(\sum_{n=1}^{20}(i^{n-1} + i^n + i^{n+1})\)
where \(i = \sqrt{-1}\)? - If \(x, y\) and \(z\) are the cube roots of unity, then what is the value of \(xy + yz + zx\)?
- If \(\alpha\), \(\beta\), \(\gamma\) are cube roots of -8, then what is \(\frac{\alpha^2 p^2 + \beta^2 q^2 + \gamma^2 r^2}{\beta^2 p^2 + \gamma^2 q^2 + \alpha^2 r^2}\) equal to ?
- What is the value of \(\frac{1}{2} + \text{Re}\left(\frac{Z_1}{Z_2}\right)\)?
- Consider the following inequalities :
I. \(1 + 4i > 3 + 2i\)
II. \(2 + 3i < 3 + 4i\)
III. \(4 + 3i > 3 + 4i\)
where \(i = \sqrt{-1}\)
How many of the above are valid ?
Important Questions from Complex Numbers
If \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\), then the value of x2 + y2 is -
If x = cos θ + i sin θ, then the value of \({x^n} + \frac{1}{{{x^n}}}\) is:
If \(\left| {\begin{array}{*{20}{c}} {6i}&{ - 3i}&1\\ 4&{3i}&{ - 1}\\ {20}&3&i \end{array}} \right| = x + iy\), then the values of x and y are:
If iz3 + z2 - z + i = 0, then the value of |z| is:
Nature of the triangle formed by the points representing the complex numbers 3 + 4i, 8 - 6i and 13 + 9i is:
