If z1, z2 are two complex numbers such that |(z1-z2)/(z1+z2)|=1 and t z1=k z2 where k ∈ℝ, then t

step 1 We are given that z1 minus z2 divide z1 plus z2 and its whole mode that is equal to 1 and iota z

If z1, z2 are two complex numbers such that...

If $z_{1}, z_{2}$ are two complex numbers such that $\left|\frac{z_{1}-z_{2}}{z_{1}+z_{2}}\right|=1$ and $t z_{1}=k z_{2}$ where $k \in \mathbb{R}$, then the angle between $\left(z_{1}-z_{2}\right)$ and $\left(z_{1}+z_{2}\right)$ is(A) $\tan ^{-1}\left(\frac{2 k}{k^{2}+1}\right)$ (B) $\tan ^{-1}\left(\frac{2 k}{1-k^{2}}\right)$ (C) $-2 \tan ^{-1}(k)$ (D) $2 \tan ^{-1}(k)$

If $z_{1}, z_{2}$ are two complex numbers such that $\left|\frac{z_{1}-z_{2}}{z_{1}+z_{2}}\right|=1$ and $t z_{1}=k z_{2}$ where $k \in \mathbb{R}$, then the angle between $\left(z_{1}-z_{2}\right)$ and $\left(z_{1}+z_{2}\right)$ is(A) $\tan ^{-1}\left(\frac{2 k}{k^{2}+1}\right)$
(B) $\tan ^{-1}\left(\frac{2 k}{1-k^{2}}\right)$
(C) $-2 \tan ^{-1}(k)$
(D) $2 \tan ^{-1}(k)$

Key Concepts

Real Scaling in Complex Relationships

When a complex number is multiplied by a real number, its magnitude is scaled without altering its direction (provided the real factor is positive). This property is used to relate complex numbers through equations like t·z? = k·z? with k ? ?, ensuring that the orientation (argument) of the complex numbers is preserved or altered in a predictable manner.

Angle Between Complex Vectors

The angle between two complex vectors is determined by the difference in their arguments. When analyzing expressions like (z? - z?) and (z? + z?), computing the angle between them involves using trigonometric relationships to express this difference, a typical approach in problems where specific angular relationships are given.

Complex Numbers

Complex numbers are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. They can be represented in a Cartesian form or in polar form (with a modulus and an argument), and this dual representation facilitates both algebraic operations and geometric interpretations in the complex plane.

Modulus and Argument

The modulus of a complex number measures its distance from the origin, while the argument is the angle it makes with the positive real axis. The modulus is crucial for understanding the size of complex numbers, and the argument is key to understanding their direction. For instance, equating the moduli of two complex expressions involves setting up relationships between distances in the complex plane.

Geometric Interpretation of Operations

Addition, subtraction, and division of complex numbers correspond to vector operations in the plane. The expressions z? - z? and z? + z? represent vectors defined by complex numbers, and comparing their magnitudes or arguments is akin to comparing the lengths or directions of these vectors in geometry.

Transcript

Please give Ace some feedback