If z = x + iy and x^2 + y^2 = 1,then frac{1 + | vedclass

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

Question

(B) Given $z = x + iy$ and $x^2 + y^2 = 1$. We know that $|z|^2 = x^2 + y^2 = 1$,so $z\bar{z} = 1$,which implies $\bar{z} = \frac{1}{z}$. Consider the

Vedclass · Accepted Answer

$z$

Vedclass · Answer

(B) Given $z = x + iy$ and $x^2 + y^2 = 1$. We know that $|z|^2 = x^2 + y^2 = 1$,so $z\bar{z} = 1$,which implies $\bar{z} = \frac{1}{z}$. Consider the expression $E = \frac{1 + x + iy}{1 + x - iy}$. Substitute $x + iy = z$ and $x - iy = \bar{z}$: $E = \frac{1 + z}{1 + \bar{z}}$. Since $\bar{z} = \frac{1}{z}$,we have: $E = \frac{1 + z}{1 + \frac{1}{z}} = \frac{1 + z}{\frac{z + 1}{z}}$. $E = \frac{(1 + z) \cdot z}{z + 1} = z$. Thus,the correct option is $B$.

If $z = x + iy$ and $x^2 + y^2 = 1$,then $\frac{1 + x + iy}{1 + x - iy} = $

Similar Questions

If ${a^2} + {b^2} = 1,$ then $\frac{{1 + b + ia}}{{1 + b - ia}} = $

If $z = 3 - 4i$,then ${z^4} - 3{z^3} + 3{z^2} + 99z - 95$ is equal to

If $e^{i \theta} = \operatorname{cis} \theta$, then find the value of $\sum_{n=0}^{\infty} \frac{\cos (n \theta)}{2^n}$.

$\frac{1 + 7i}{(2 - i)^2} = $

If $z = \frac{\sqrt{3}}{2} + \frac{i}{2}$ where $i = \sqrt{-1}$,then $(1 + iz + z^5 + iz^8)^9$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module