\( z_{1}, z_{2}, z_{3} \) represents distinct points \( A, B, C \) in Argand
Plane.
\begin{tabul....
\( z_{1}, z_{2}, z_{3} \) represents distinct points \( A, B, C \) in Argand
Plane.
\begin{tabular}{|c|c|c|c|c|c|}
\hline \multicolumn{2}{|r|}{ Column-I } & \multicolumn{2}{|c|}{ Column-II } & & \\
\hline (A) & \( 2 z_{1} z_{3}=z_{1} z_{2}+z_{2} z_{3} \) & (P) & \begin{tabular}{l}
\( A, B, C \) are \\
collinear
\end{tabular} & & \\
\hline (B) & \begin{tabular}{l}
\( \left|z_{3}-z_{1}\right|^{2}+\left|z_{3}-z_{2}\right|^{2} \) \\
\( =\left|z_{1}-z_{2}\right|^{2} \)
\end{tabular} & (Q) & \begin{tabular}{l}
\( A, B, C \) lie \\
on circle \\
whose \\
diameter \\
is \( A B \)
\end{tabular} & & \\
\hline (C) & \begin{tabular}{l}
\( \lambda_{1}, \lambda_{2}, \lambda_{3} \in \) real, \( \lambda_{1} z_{1}+ \) \\
\( \lambda_{2} z_{2}+\lambda_{3} z_{3}=0, \lambda_{1}+\lambda_{2}+ \) \\
\( \lambda_{3}=0 \)
\end{tabular} & (R) & \begin{tabular}{l}
\( A, B, C \) lie \\
in circle \\
which \\
passes \\
through \\
\( \mathrm{O}(0,0) \)
\end{tabular} & & \\
\hline (D) & \begin{tabular}{l}
\( \frac{1}{z_{1}-z_{2}}+\frac{1}{z_{2}-z_{3}} \) \\
\( +\frac{1}{z_{3}-z_{1}}=0 \)
\end{tabular} & (S) & \begin{tabular}{l}
\( A, \quad B, \quad C \) \\
make \\
equilateral \\
triangle
\end{tabular} & \begin{tabular}{l}
\( (1) \) \\
\( (2) \) \\
\( (3) \) \\
\( (4) \)
\end{tabular} & \begin{tabular}{l}
A-P; B-Q; C-R; D-S \\
A-P; B-R; C-S; D-S \\
A-R; B-Q; C-P; D-S \\
A-P;B-R; C-S; D-Q
\end{tabular} \\
\hline
\end{tabular}
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