Question

If one vertex of a square whose diagonals intersect at the origin is $3(cos\theta +isin\theta )$, then find the two adjacent vertices.

• A. vertices B and D are represented by $\pm 3(sin\theta -icos\theta )$
• B. vertices B and D are represented by $\pm 3(sin\theta +icos\theta )$
• C. vertices B and D are represented by $\pm 3(cos\theta -isin\theta )$
• D. vertices B and D are represented by $\pm 3(cos\theta +isin\theta )$

Answer 1

The correct option is A vertices B and D are represented by $\pm 3(sin\theta -icos\theta )$

Let the vertex A be $3(cos\theta +isin\theta )$, then OB and OD
can be obtained by rotating OA through $\pi /2$ and $-\pi /2$. Thus,
$\overline{OB}=\left(\overline{OA}\right)e^{i\pi /2}$ and $\overline{OD}=\overline{OA}{e}^{-i\pi /2}$
$⟹\overline{OB}=3(cos\theta +isin\theta )i$ and $\overline{OD}=3(cos\theta +isin)(-i)$
$⟹\overline{OB}=3(-sin\theta +icos\theta )$ and $\overline{OD}=3(sin\theta -icos\theta )$
Thus, vertices B and D are represented by $\pm 3(sin\theta -icos\theta )$
Ans: A