Question

If ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ are three non-zero complex numbers such that ${{z}_{3}}=\left( 1-\lambda \right){{z}_{1}}+\lambda {{z}_{2}}$, where $\lambda \in R-\left\{ 0 \right\}$, then determine the curve on which the points ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ lies.

Answer 1

Hint: We start solving the problem by making the arrangements in the given relation between the points ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ . We then use the facts that if $z$ and $z'$ are two complex numbers, then $z-z'$ is the distance between those two points on the complex plane and if three points A, B, C are collinear, then $AB=\alpha AC$. We use these two facts for the relation obtained between the distances of the points ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ to get the curve where they were lying.

Complete step-by-step solution
 According to the problem, we are given that ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ are three non-zero complex numbers such that ${{z}_{3}}=\left( 1-\lambda \right){{z}_{1}}+\lambda {{z}_{2}}$, where $\lambda \in R-\left\{ 0 \right\}$. We need to find the curve on which the points ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ lies.
We have ${{z}_{3}}=\left( 1-\lambda \right){{z}_{1}}+\lambda {{z}_{2}}$.
$\Rightarrow {{z}_{3}}={{z}_{1}}-\lambda {{z}_{1}}+\lambda {{z}_{2}}$.
$\Rightarrow {{z}_{3}}-{{z}_{1}}=\lambda {{z}_{2}}-\lambda {{z}_{1}}$.
$\Rightarrow {{z}_{3}}-{{z}_{1}}=\lambda \left( {{z}_{2}}-{{z}_{1}} \right)$ ---(1).
We know that if $z$ and $z'$ are two complex numbers, then $z-z'$ is the distance between those two points on the complex plane.
So from equation (1), we have found that the distance between the points ${{z}_{1}}$ and ${{z}_{3}}$ is $\lambda $ times the distance between the points ${{z}_{2}}$ and ${{z}_{1}}$.
We know that if three points A, B, C are collinear, then $AB=\alpha AC$ which is clearly satisfied by the points ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$.
So, the points ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ are collinear.
∴ The points ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ lie on a straight line.

Note: Whenever we get this type of problems, we try to establish a relation between the given points and check whether they were satisfied by any of the curves. We should know that the properties satisfied in real plane are also satisfied in the complex plane. We can also use the formula to find the points that lie between two points A and B as $\left( 1-\alpha \right)A+\alpha B$ where $0\le \alpha \le 1$. Similarly, we can expect problems to find the curve satisfying the point ${{z}_{1}}$, ${{z}_{2}}$ if they satisfies ${{\left| {{z}_{1}} \right|}^{2}}={{z}_{2}}\overline{{{z}_{1}}}$.