Question

$z_1,z_2,z_3$ and $z_1^{\prime },z_2^{\prime },z_3^{\prime }$ are nonzero complex numbers such that $z_3=(1-\lambda )z_1+\lambda z_2$ and $z_3^{\prime }=(1-\mu )z_1^{\prime }+\mu z_2^{\prime }$, then which of the following statements is/are true?

• A. If $\lambda ,\mu ϵR-\left\{0\right\}$, then $z_1,z_2,$ and $z_3$ are collinear and $z_1^{\prime },z_2^{\prime },z_3^{\prime }$ are collinear separately.
• B. If $\lambda ,\mu$ are complex numbers, where $\lambda =\mu$, then triangles formed by points $z_1,z_2,z_3$ and $z_1^{\prime },z_2^{\prime },z_3^{\prime }$ are similar.
• C. If $\lambda ,mu$ are distinct complex numbers, then points $z_1,z_2,z_3$ and $z_1^{\prime },z_2^{\prime },z_3^{\prime }$ are not connected by any well defined geometry.
• D. If $0<\lambda <1$, then $z_3$ divides the line joining $z_1$ and $z_2$ internally and if $\mu >1$, then $z_3^{\prime }$ divides the line joining of $z_1^{\prime },z_2^{\prime }$ externally.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A If $\lambda ,\mu ϵR-\left\{0\right\}$, then $z_1,z_2,$ and $z_3$ are collinear and $z_1^{\prime },z_2^{\prime },z_3^{\prime }$ are collinear separately.
B If $\lambda ,\mu$ are complex numbers, where $\lambda =\mu$, then triangles formed by points $z_1,z_2,z_3$ and $z_1^{\prime },z_2^{\prime },z_3^{\prime }$ are similar.
C If $\lambda ,mu$ are distinct complex numbers, then points $z_1,z_2,z_3$ and $z_1^{\prime },z_2^{\prime },z_3^{\prime }$ are not connected by any well defined geometry.
D If $0<\lambda <1$, then $z_3$ divides the line joining $z_1$ and $z_2$ internally and if $\mu >1$, then $z_3^{\prime }$ divides the line joining of $z_1^{\prime },z_2^{\prime }$ externally.

$z_3=(1-\lambda )z_1+z_2=\frac{(1-\lambda )z_1+\lambda z_2}{1-\lambda +\lambda }$
Hence, $z_3$ divides the line joining $A\left(z_1\right)$ and $B\left(z_2\right)$ in the ratio $\lambda :(1-\lambda )$.
That means the given points are collinear. also, the ratio $\lambda /(1-\lambda )>$ (or $0<\lambda <1)$ if $z_3$ divides the line joining $z_1$ and $z_2$ internally and $\mu /(1-\mu )<0$ (or $\mu <0$ or $\mu >1)$
if $z_3^{\prime }$ divides the line joining $z_1^{\prime },z_2^{\prime }$ externally.
When $\lambda ,\mu$ are complex numbers, where $\lambda =\mu$, we have $z_3=(1-\lambda )z_1+\lambda z_2$ and $z_3^{\prime }=(1-\lambda )z_1^{\prime }+\lambda z_2^{\prime }$.
Comparing the value of $\lambda$, we have
$\frac{z_3-z_1}{z_2-z_1}=\frac{z_3^{\prime }-z_1^{\prime }}{z_2^{\prime }-z_1^{\prime }}$
$\Rightarrow \left|\frac{z_3-z_1}{z_2-z_1}\right|=\left|\frac{z_3^{\prime }-z_1^{\prime }}{z_2^{\prime }-z_1^{\prime }}\right|$ and $arg\left(\frac{z_3-z_1}{z_2-z_1}\right)=arg\left(\frac{z_3^{\prime }-z_1^{\prime }}{z_2^{\prime }-z_1^{\prime }}\right)$
$\Rightarrow \frac{AC}{AB}=\frac{PR}{PQ}$ and $\angle BAC=\angle QPR$
Hence, triangles ABC and PQR are similar.