Complex Numbers 3 Question 7
7.
If $a, b, c$ and $u, v, w$ are the complex numbers representing the vertices of two triangles such that $c=(1-r) a+r b$ and $w=(1-r) u+r v$, where $r$ is a complex number, then the two triangles
$(1985,2 M)$
(a) have the same area
(b) are similar
(c) are congruent
(d) None of these
Show Answer
Answer:
Correct Answer: 7. (b)
Solution:
- Since $a, b, c$ and $u, v, w$ are the vertices of two triangles.
Also, $\quad c=(1-r) a+r b $
and $\quad w=(1-r) u+r v $
consider $\quad \begin{vmatrix}a & u & 1\\ b & v & 1\\ c & w & 1\end{vmatrix}$
Applying $R_3 \rightarrow R_3 -[(1-r) R_1 +rR_2]$
=$\begin{vmatrix} a & u & 1 \\ b & v & 1 \\ c-(1-r) a-r b & w-(1-r) u-r v & 1-(1-r)-r \end{vmatrix}$
$ \begin{aligned} & =\begin{vmatrix} a & u & 1 \\ b & v & 1 \\ 0 & 0 & 0 \end{vmatrix}=0 \end{aligned} $
