If $z_1, z_2$ are two distinct complex numbers such that \[ \frac{|z_1 - 2z_2|}{\left| \frac{1}{2} - z_1 \overline{z_2} \right|} = 2, \] then
- either $z_1$ lies on a circle of radius 1 or $z_2$ lies on a circle of radius $\frac{1}{2}$.
- either $z_1$ lies on a circle of radius $\frac{1}{2}$ or $z_2$ lies on a circle of radius 1.
- $z_1$ lies on a circle of radius $\frac{1}{2}$ and $z_2$ lies on a circle of radius 1.
- both $z_1$ and $z_2$ lie on the same circle.
The Correct Option is A
Approach Solution - 1
To solve the given problem, we start by examining the equation:
\(\frac{|z_1 - 2z_2|}{\left| \frac{1}{2} - z_1 \overline{z_2} \right|} = 2\)
This equation essentially represents a condition involving the distances between two points on the complex plane.
First, let's simplify the equation:
\(|z_1 - 2z_2| = 2 \left| \frac{1}{2} - z_1 \overline{z_2} \right|\)
Consider \(|z_1 - 2z_2|\)as the distance from \(z_1\) to \(2z_2\):
- If we assume that \(z_1\) lies on a circle of radius 1, then we have:
\(|z_1| = 1 \ \text{or} \ |z_2 - 0| = \frac{1}{2}\)
Similarly, consider the absolute value:
\(\left| \frac{1}{2} - z_1 \overline{z_2} \right|\)represents the modulus of a complex number.
- If we assume \(z_2\) lies on a circle of radius \(\frac{1}{2}\), this satisfies the equation provided since both sides could balance out to remain equal.
Therefore, the correct interpretation is that either:
- \(z_1\) lies on a circle of radius 1.
- or \(z_2\) lies on a circle of radius \(\frac{1}{2}\).
Conclusion: The correct answer is that either \(z_1\) lies on a circle of radius 1 or \(z_2\) lies on a circle of radius \(\frac{1}{2}\).
Approach Solution -2
\[ \frac{z_1 - 2z_2}{\frac{1}{2} - 2z_1z_2} \times \frac{\overline{z_1} - 2\overline{z_2}}{\frac{1}{2} - z_1z_2} = 4 \]
\[ \lvert z_1 \rvert^2 \left\lvert 2z_1z_2 - 2z_2\overline{z_1} + 4\lvert z_2 \rvert^2 \right\rvert^2 \]
\[ = 4 \left( \frac{1}{4}(z_1\overline{z_2} - z_2\overline{z_1})^2 + \lvert z_1 \rvert^2 \lvert z_2 \rvert^2 \right) \]
\[ z_1\overline{z_1} + 2z_2 \cdot 2\overline{z_2} - z_1z_2 \cdot z_2\overline{z_2} - 1 = 0 \]
\[ (z_1\overline{z_1} - 1)(1 - 2z_2\overline{z_2}) = 0 \]
\[ (\lvert z_1 \rvert^2 - 1)\left((2\lvert z_2 \rvert^2 - 1)\right) = 0 \]
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Complex numbers Questions
- If \(z = x + iy, \quad xy \neq 0\) satisfies the equation \(z^2 + iz = 0\), then \(|z^2|\); equal to
- If \(r = |z|,\ θ = arg(z)\) and \( z = 2 – 2\ tan (\frac {5\pi}{8})\) then find \((r, θ)\).
- Let \(z_1\) and \(z_2\) be two complex numbers such that \(z_1 + z_2 = 5\) and \(z_1^3+z_2^3 =20 +15i\), then the value of \(| z_1^4+z_2^4 |\) is equal to
- If \( z = x + iy \), \( xy \neq 0 \), satisfies the equation \( z^2 + i\overline{z} = 0 \), then \( |z|^2 \) is equal to:
- The values of \(\alpha\), for which
lie in the interval
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.