Question

Find the value of $k$, if for the complex numbers $z_{1}$ and $z_{2}$, $\left|1-\bar{z}_{1}z_{2}\right|^{2}-\left|z_{1}-z_{2}\right|^{2} =k\left(1-\left|z_{1}\right|^{2}\right)\left(1-\left|z_{2}\right|^{2}\right)$.

• A. $2$
• B. $3$
• C. $1$
• D. $4$

Answer 1

The Correct Option is C

$L.H.S.$ $=\left|1-\bar{z}_{1}z_{2}\right|^{2}-\left|z_{1}-z_{2}\right|^{2}$ $=\left(1-\bar{z}_{1}z_{2}\right)\left(\overline{1-\bar{z}_{1} z}_{2}\right)-\left(z_{1}-z_{2}\right)\left(\overline{z_{1}-z_{2}}\right)$ $=\left(1-\bar{z}_{1}z_{2}\right)\left(1-z_{1}\bar{z}_{2}\right)-\left(z_{1}-z_{2}\right)\left(\bar{z}_{1}-\bar{z}_{2}\right)$ $=1+z_{1}\bar{z}_{1} z_{2} \bar{z}_{2}-z_{1} \bar{z}_{1}-z_{2} \bar{z}_{2}$ $=1+\left|z_{1}\right|^{2}\cdot\left|z_{2}\right|^{2}-\left|z_{1}\right|^{2}-\left|z_{2}\right|^{2}$ $=\left(1-\left|z_{1}\right|^{2}\right)\left(1-\left|z_{2}\right|^{2}\right)$ $R.H.S. =k\left(1-\left|z_{1}\right|^{2}\right)\left(1-\left|z_{2}\right|^{2}\right)$ Hence, equating $L.H.S.$ and $R.H.S.$, we get $k = 1$.