A) \[{{z}_{1}}^{2}+{{z}_{2}}^{2}+{{z}_{3}}^{2}={{z}_{1}}{{z}_{2}}{{z}_{3}}\]
B) \[{{({{z}_{3}}-{{z}_{1}})}^{2}}={{z}_{3}}-{{z}_{2}}\]
C) \[{{({{z}_{1}}-{{z}_{2}})}^{2}}=({{z}_{1}}-{{z}_{3}})\,({{z}_{3}}-{{z}_{2}})\]
D) \[{{({{z}_{1}}-{{z}_{2}})}^{2}}=2({{z}_{1}}-{{z}_{3}})\,({{z}_{3}}-{{z}_{2}})\]
Correct Answer: D
Solution :
\[BC=AC\] and \[\angle C=\pi /2\] By rotation about \[C\] in anti-clockwise sense \[CB=CA\,{{e}^{i\,\pi /2}}\] Þ \[({{z}_{2}}-{{z}_{3}})=({{z}_{1}}-{{z}_{3}})\,{{e}^{i\,\pi /2}}=i\,({{z}_{1}}-{{z}_{3}})\] Þ \[{{({{z}_{2}}-{{z}_{3}})}^{2}}=-{{({{z}_{1}}-{{z}_{3}})}^{2}}\] Þ \[z_{2}^{2}+z_{3}^{2}-2{{z}_{2}}{{z}_{3}}=-z_{1}^{2}-z_{3}^{2}+2{{z}_{1}}{{z}_{3}}\] Þ \[z_{1}^{2}+z_{2}^{2}-2{{z}_{1}}{{z}_{2}}=2{{z}_{1}}{{z}_{3}}+2{{z}_{2}}{{z}_{3}}-2z_{3}^{2}-2{{z}_{1}}{{z}_{2}}\] Þ \[{{({{z}_{1}}-{{z}_{2}})}^{2}}=2\,[({{z}_{1}}{{z}_{3}}-z_{3}^{2})-({{z}_{1}}{{z}_{2}}-{{z}_{2}}{{z}_{3}})]\] Þ \[{{({{z}_{1}}-{{z}_{2}})}^{2}}=2({{z}_{1}}-{{z}_{3}})\,({{z}_{3}}-{{z}_{2}})\].You need to login to perform this action.
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