Menu Close

Passage-If-z-1-z-2-and-z-3-are-three-complex-numbers-representing-the-points-A-B-and-C-respectively-in-the-Argands-plane-and-BAC-then-z-3-z-1-z-2-z-1-AC-




Question Number 140260 by EnterUsername last updated on 05/May/21
Passage: If z_1 , z_2  and z_3  are three complex numbers  representing the points A, B and C, respectively, in the  Argands plane and ∠BAC=α, then                      ((z_3 −z_1 )/(z_2 −z_1 ))=(((AC)/(AB)))(cosα+isinα)  (i) If the roots of the equation                          z^3 +3a_1 z^2 +3a_2 z+a_3 =0         represent the vertices of an equilateral triangle, then         (A) a_1 ^2 =a_3                                        (B) a_1 ^2 =a_2                (C) a_1 ^2 =a_2 a_3                                      (D) a_1 ^3 =a_2 a_3
$$\mathrm{Passage}:\:\mathrm{If}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} \:\mathrm{and}\:{z}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{three}\:\mathrm{complex}\:\mathrm{numbers} \\ $$$$\mathrm{representing}\:\mathrm{the}\:\mathrm{points}\:{A},\:{B}\:\mathrm{and}\:{C},\:\mathrm{respectively},\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{Argands}\:\mathrm{plane}\:\mathrm{and}\:\angle{BAC}=\alpha,\:\mathrm{then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{z}_{\mathrm{3}} −{z}_{\mathrm{1}} }{{z}_{\mathrm{2}} −{z}_{\mathrm{1}} }=\left(\frac{{AC}}{{AB}}\right)\left(\mathrm{cos}\alpha+{i}\mathrm{sin}\alpha\right) \\ $$$$\left({i}\right)\:\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{z}^{\mathrm{3}} +\mathrm{3}{a}_{\mathrm{1}} {z}^{\mathrm{2}} +\mathrm{3}{a}_{\mathrm{2}} {z}+{a}_{\mathrm{3}} =\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\mathrm{represent}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle},\:\mathrm{then} \\ $$$$\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:{a}_{\mathrm{1}} ^{\mathrm{2}} ={a}_{\mathrm{3}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:{a}_{\mathrm{1}} ^{\mathrm{2}} ={a}_{\mathrm{2}} \:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:{a}_{\mathrm{1}} ^{\mathrm{2}} ={a}_{\mathrm{2}} {a}_{\mathrm{3}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:{a}_{\mathrm{1}} ^{\mathrm{3}} ={a}_{\mathrm{2}} {a}_{\mathrm{3}} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *