Question

Let $A$ and $B$ be two distinct points denoting the complex numbers $\alpha$ and $\beta$ respectively. A complex number $z$ lies between $A$ and $B$ where $z\ne \alpha ,z\ne \beta$. Which of the following relation(s) hold good?

• A. $|\alpha -z|+|z-\beta |=|\alpha -\beta |$
• B. $\exists$ a positive real number ' $t$ ' such that $z=(1-t)\alpha +t\beta$
• C. $\left|\begin{array}{cc}z-\alpha & \overline{z}-\bar{\alpha }\\ \beta -\alpha & \bar{\beta }-\bar{\alpha }\end{array}\right|=0$
• D. $\left|\begin{array}{ccc}z& \overline{z}& 1\\ \alpha & \bar{\alpha }& 1\\ \beta & \bar{\beta }& 1\end{array}\right|=0$

Answer 1

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

$AP+PB=AB$
$$\begin{gathered} |z-\alpha |+|\beta -z|=|\beta -\alpha |\Rightarrow \text{ A True } \\ <br/>\text{ Now }z=\alpha +t(\beta -\alpha ) \end{gathered}$$
$=(1-t)\alpha +t\beta \text{ where }t\in (0,1)\Rightarrow B\text{ is True }$
$\text{ again }\frac{z-\alpha }{\beta -\alpha }\text{ is real }\Rightarrow \frac{z-\alpha }{\beta -\alpha }=\frac{\overline{z}-\bar{\alpha }}{\bar{\beta }-\bar{\alpha }}$
$\Rightarrow \left|\begin{array}{cc}z-\alpha & \bar{z}-\bar{\alpha }\\ \beta -\alpha & \bar{\beta }-\bar{\alpha }\end{array}\right|=0$
$\text{ also }\left|\begin{array}{ccc}z& \bar{z}& 1\\ \alpha & \bar{\alpha }& 1\\ \beta & \bar{\beta }& 1\end{array}\right|=0\text{ if and only if }\left|\begin{array}{ccc}z-\alpha & \bar{z}-\bar{\alpha }& 0\\ \alpha & \bar{\alpha }& 1\\ \beta -\alpha & \bar{\beta }-\bar{\alpha }& 0\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{cc}z-\alpha )& \bar{z}-\bar{\alpha }\\ \beta -\bar{\alpha }& \bar{\beta }-\bar{\alpha }\end{array}\right|=0$