Mathematics - Complex Number Question with Solution | TestHub

Assertion (A) is true as $z_1+z_2+z_3=0$ for distinct $z_k$ on the unit circle implies the origin is the centroid, hence an equilateral triangle. Reason (R) is also true, as $z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$ is a known condition for an equilateral triangle. To show R explains A, from $z_1+z_2+z_3=0$, we have $z_1^2+z_2^2+z_3^2=-2(z_1{z}_2+z_2{z}_3+z_3{z}_1)$. Also, $|z_k|=1⟹\overline{z_k}=1/z_k$. Thus, $z_1+z_2+z_3=0⟹\overline{z_1}+\overline{z_2}+\overline{z_3}=0⟹1/z_1+1/z_2+1/z_3=0⟹z_1{z}_2+z_2{z}_3+z_3{z}_1=0$.

Substituting this into the previous equation, $z_1^2+z_2^2+z_3^2=0$. Since both sides of R's condition ($z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$) become $0$, it is satisfied. Hence, R explains A.