What is \(\left| \frac{Z_1 + Z_2}{Z_1 - Z_2} \right|\) equal to ?
Complex Number Magnitude Calculation
The problem asks for the value of \(\left| \frac{Z_1 + Z_2}{Z_1 - Z_2} \right|\) given that \(\frac{3Z_1}{4Z_2}\) is purely imaginary.
Condition Analysis
A complex number \(w\) is purely imaginary if its real part is zero, which implies \(w = - \bar{w}\) (for \(w \neq 0\)).
Given that \(\frac{3Z_1}{4Z_2}\) is purely imaginary:
\( \frac{3Z_1}{4Z_2} = - \overline{\left(\frac{3Z_1}{4Z_2}\right)} \) \( \frac{3Z_1}{4Z_2} = - \frac{3\bar{Z_1}}{4\bar{Z_2}} \)Simplifying this equation by canceling out the common terms \(\frac{3}{4}\):
\( \frac{Z_1}{Z_2} = - \frac{\bar{Z_1}}{\bar{Z_2}} \)This can be rewritten as:
\( \frac{Z_1}{Z_2} = - \overline{\left(\frac{Z_1}{Z_2}\right)} \)This shows that the ratio \(\frac{Z_1}{Z_2}\) is itself a purely imaginary number. Let's represent this ratio as \(ik\), where \(k\) is a non-zero real number.
\( \frac{Z_1}{Z_2} = ik, \quad k \in \mathbb{R}, k \neq 0 \)Magnitude Calculation
Now, we need to find the magnitude of the expression \(\frac{Z_1 + Z_2}{Z_1 - Z_2}\). Divide both the numerator and the denominator by \(Z_2\) (assuming \(Z_2 \neq 0\)):
\( \frac{Z_1 + Z_2}{Z_1 - Z_2} = \frac{\frac{Z_1}{Z_2} + \frac{Z_2}{Z_2}}{\frac{Z_1}{Z_2} - \frac{Z_2}{Z_2}} = \frac{\frac{Z_1}{Z_2} + 1}{\frac{Z_1}{Z_2} - 1} \)Substitute \(\frac{Z_1}{Z_2} = ik\) into the expression:
\( \frac{ik + 1}{ik - 1} = \frac{1 + ik}{-1 + ik} \)Finally, calculate the magnitude of this complex number:
\( \left| \frac{1 + ik}{-1 + ik} \right| = \frac{|1 + ik|}{|-1 + ik|} \)The magnitude of a complex number \(a+bi\) is \(\sqrt{a^2 + b^2}\). Therefore:
\( \frac{|1 + ik|}{|-1 + ik|} = \frac{\sqrt{1^2 + k^2}}{\sqrt{(-1)^2 + k^2}} = \frac{\sqrt{1 + k^2}}{\sqrt{1 + k^2}} \) \( \frac{\sqrt{1 + k^2}}{\sqrt{1 + k^2}} = 1 \)Thus, the magnitude \(\left| \frac{Z_1 + Z_2}{Z_1 - Z_2} \right|\) is equal to 1.
Similar Questions
If
\(\left( \frac{1-i}{1+i} \right)^{2m} \left( \frac{1+i}{1-i} \right)^{2n} = 1\)
where \(i = \sqrt{-1}\), then what is the smallest positive value of \((m – n)\)?
- 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 A2 + B2 + C2 = 0, then what is the value of the following?
\(\Delta = \begin{vmatrix} 1 & \cos C & \cos B \\\ \cos C & 1 & \cos A \\\ \cos B & \cos A & 1 \end{vmatrix}\)If ω is a non-real cube root of unity, then what is a root of the following equation?
Important Questions from Complex Numbers
If A + iB = tan (x + iy), then the value of tan 2x is?
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:
