For any two non-zero complex numbers z_1 and | vedclass

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

Question

(A) We are given $|z_1+z_2|^2 = |z_1|^2 + |z_2|^2$. Using the property $|z|^2 = z \bar{z}$, we can write: $(z_1+z_2)(\bar{z_1}+\bar{z_2}) = z_1 \bar{z

Vedclass · Accepted Answer

$\operatorname{Re}\left(\frac{z_1}{z_2}\right)=0$

Vedclass · Answer

(A) We are given $|z_1+z_2|^2 = |z_1|^2 + |z_2|^2$. Using the property $|z|^2 = z \bar{z}$, we can write: $(z_1+z_2)(\bar{z_1}+\bar{z_2}) = z_1 \bar{z_1} + z_2 \bar{z_2}$. Expanding the left side: $z_1 \bar{z_1} + z_1 \bar{z_2} + z_2 \bar{z_1} + z_2 \bar{z_2} = z_1 \bar{z_1} + z_2 \bar{z_2}$. Subtracting $|z_1|^2$ and $|z_2|^2$ from both sides, we get: $z_1 \bar{z_2} + z_2 \bar{z_1} = 0$. This can be written as $z_1 \bar{z_2} + \overline{z_1 \bar{z_2}} = 0$. Since $z + \bar{z} = 2 \operatorname{Re}(z)$, we have $2 \operatorname{Re}(z_1 \bar{z_2}) = 0$, which implies $\operatorname{Re}(z_1 \bar{z_2}) = 0$. Dividing by $|z_2|^2$, we get $\operatorname{Re}\left(\frac{z_1 \bar{z_2}}{z_2 \bar{z_2}}\right) = 0$, which simplifies to $\operatorname{Re}\left(\frac{z_1}{z_2}\right) = 0$.

For any two non-zero complex numbers $z_1$ and $z_2$, if $|z_1+z_2|^2=|z_1|^2+|z_2|^2$, then

Similar Questions

If $P, Q, R, S$ are represented by the complex numbers $4 + i, 1 + 6i, -4 + 3i, -1 - 2i$ respectively,then $PQRS$ is a

$A$ man walks a distance of $3$ units from the origin towards the north-east $(N 45^{\circ} E)$ direction. From there,he walks a distance of $4$ units towards the north-west $(N 45^{\circ} W)$ direction to reach a point $P$. Then the position of $P$ in the Argand plane is

If $a = \operatorname{Im}\left(\frac{1+z^2}{2iz}\right)$ and $z$ is any non-zero complex number such that $|z|=1$,then $a=$

$P$ is a point denoting $z$ in the Argand diagram. If $\frac{z-i}{z-1}$ is always purely imaginary,then the locus of $P$ is

The maximum value of $|z|$ where $z$ satisfies the condition $\left| z + \frac{2}{z} \right| = 2$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module