Complex Numbers And Quadratic Equations question 327
Question: The complex numbers $ z_1,z_2,z_3 $ are the vertices of a triangle. Then the complex numbers $ z $ which make the triangle into a parallelogram is
Options:
A) $ z_1+z_2-z_3 $
B) $ z_1-z_2+z_3 $
C) $ z_2+z_3-z_1 $
D) All the above
Show Answer
Answer:
Correct Answer: D
Solution:
Let $ A,B,C $ be the points represented by the numbers $ z_1,z_2,z_3 $ and P be the point represented by $ z $ Now the four points $ A,B,C,P $ form a parallelogram in the following three orders. (i) $ A,B,P,C $ (ii) $ B,C,P,A $ and (iii) $ C,A,P,B $ In case (i), the condition for $ A,B,P,C $ to form a parallelogram is $ \overrightarrow{AB}=\overrightarrow{CP} $ i.e., $ z_2-z_1=z-z_3 $ or $ z=z_2+z_3-z_1 $ Similarly in case (ii) and (iii), the required points $ \overrightarrow{BC}=\overrightarrow{AP} $ or $ z_3-z_2=z-z_1 $ i.e., $ z=z_3+z_1-z_2 $ and $ \overrightarrow{CA}=\overrightarrow{BP} $ or $ z_1-z_3=z-z_2 $ i.e., $ z=z_1+z_2-z_3 $
BETA
