For a, b, c, d in R, if z_1 = a + ib and z_2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given $|z_1| = |z_2| = 1$, we can write $z_1 = \cos \alpha + i \sin \alpha$ and $z_2 = \cos \beta + i \sin \beta$.$\operatorname{Re}(z_1 \bar{z}_2

Vedclass · Accepted Answer

$\operatorname{Re}(w_1 \bar{w}_2) = 0$

Vedclass · Answer

(A) Given $|z_1| = |z_2| = 1$, we can write $z_1 = \cos \alpha + i \sin \alpha$ and $z_2 = \cos \beta + i \sin \beta$.$\operatorname{Re}(z_1 \bar{z}_2) = ac + bd = \cos \alpha \cos \beta + \sin \alpha \sin \beta = \cos(\alpha - \beta) = 0$.This implies $\alpha - \beta = \pm \frac{\pi}{2}$.Now, for $w_1 = a + ic = \cos \alpha + i \cos \beta$ and $w_2 = b + id = \sin \alpha + i \sin \beta$, we calculate $\operatorname{Re}(w_1 \bar{w}_2) = ab + cd$.$\operatorname{Re}(w_1 \bar{w}_2) = \cos \alpha \sin \alpha + \cos \beta \sin \beta = \frac{1}{2}(\sin 2\alpha + \sin 2\beta)$.Using the sum-to-product formula, $\frac{1}{2}(2 \sin(\alpha + \beta) \cos(\alpha - \beta))$.Since $\cos(\alpha - \beta) = \cos(\pm \frac{\pi}{2}) = 0$, the expression equals $0$.Thus, $\operatorname{Re}(w_1 \bar{w}_2) = 0$.

For $a, b, c, d \in R$, if $z_1 = a + ib$ and $z_2 = c + id$ are such that $|z_1| = |z_2| = 1$ and $\operatorname{Re}(z_1 \bar{z}_2) = 0$, then the pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfy

Similar Questions

If $P$ is the point in the Argand diagram corresponding to the complex number $\sqrt{3}+i$ and if $OPQ$ is an isosceles right-angled triangle,right-angled at $O$,then $Q$ represents the complex number

The largest value of $r$ for which the region represented by the set $\{ \omega \in \mathbb{C} : |\omega - 4 - i| \le r \}$ is contained in the region represented by the set $\{ z \in \mathbb{C} : |z - 1| \le |z + i| \}$ is equal to

The real part of ${e^{e^{i\theta }}}$ is

If $x + iy = \sqrt{\phi + i\psi}$,where $i = \sqrt{-1}$ and $\phi$ and $\psi$ are non-zero real parameters,then $\phi = \text{constant}$ and $\psi = \text{constant}$ represent two systems of rectangular hyperbolas which intersect at an angle of:

Let $z=x+iy$ be a complex number with $x, y \in \mathbb{Z}$. Then,the area (in sq units) of the rectangle whose vertices are the roots of the equation $\bar{z} \cdot z^3+z \cdot \bar{z}^3=350$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module