- Home
- Standard 11
- Mathematics
4-1.Complex numbers
normal
If $z_1, z_2, z_3$ $\in$ $C$ such that $|z_1| = |z_2| = |z_3| = 2$, then greatest value of expression $|z_1 - z_2|.|z_2 - z_3| + |z_3 - z_1|.|z_1 - z_2| + |z_2 - z_3||z_3 - z_1|$ is
A
$18$
B
$36$
C
$9$
D
$72$
Solution
Let $\left|z_{1}-z_{2}\right|=\alpha,\left|z_{2}-z_{3}\right|=\beta ;\left|z_{3}-z_{1}\right|=\gamma$
$\alpha \beta+\beta \gamma+\gamma \alpha<\alpha^{2}+\beta^{2}+\gamma^{2}$
$=3\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)-\left|z_{1}+z_{2}+z_{3}\right|^{2}<3\left(\left|z_{1}\right|^{2}\right.$
$\left.+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)$
$\Rightarrow \alpha \beta+\beta \gamma+\gamma \alpha \leq 36$
Standard 11
Mathematics
