Menu Close

Let-z-1-and-z-2-be-two-distinct-complex-numbers-and-let-z-1-t-z-1-tz-2-for-some-real-number-t-with-0-lt-t-lt-1-If-arg-w-denotes-the-principal-argument-of-a-non-zero-complex-number-w-




Question Number 21006 by Tinkutara last updated on 10/Sep/17
Let z_1  and z_2  be two distinct complex  numbers and let z = (1 − t)z_1  + tz_2  for  some real number t with 0 < t < 1. If  arg(w) denotes the principal argument  of a non-zero complex number w, then  (1) ∣z − z_1 ∣ + ∣z − z_2 ∣ = ∣z_1  − z_2 ∣  (2) Arg (z − z_1 ) = Arg (z − z_2 )  (3)  determinant (((z − z_1 ),(z^�  − z_1 ^� )),((z_2  − z_1 ),(z_2 ^�  − z_1 ^� ))) = 0  (4) Arg (z − z_1 ) = Arg (z_2  − z_1 )
$$\mathrm{Let}\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{complex} \\ $$$$\mathrm{numbers}\:\mathrm{and}\:\mathrm{let}\:{z}\:=\:\left(\mathrm{1}\:−\:{t}\right){z}_{\mathrm{1}} \:+\:{tz}_{\mathrm{2}} \:\mathrm{for} \\ $$$$\mathrm{some}\:\mathrm{real}\:\mathrm{number}\:{t}\:\mathrm{with}\:\mathrm{0}\:<\:{t}\:<\:\mathrm{1}.\:\mathrm{If} \\ $$$$\mathrm{arg}\left({w}\right)\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{principal}\:\mathrm{argument} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{non}-\mathrm{zero}\:\mathrm{complex}\:\mathrm{number}\:{w},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:\mid{z}\:−\:{z}_{\mathrm{1}} \mid\:+\:\mid{z}\:−\:{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{1}} \:−\:{z}_{\mathrm{2}} \mid \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Arg}\:\left({z}\:−\:{z}_{\mathrm{1}} \right)\:=\:\mathrm{Arg}\:\left({z}\:−\:{z}_{\mathrm{2}} \right) \\ $$$$\left(\mathrm{3}\right)\:\begin{vmatrix}{{z}\:−\:{z}_{\mathrm{1}} }&{\bar {{z}}\:−\:\bar {{z}}_{\mathrm{1}} }\\{{z}_{\mathrm{2}} \:−\:{z}_{\mathrm{1}} }&{\bar {{z}}_{\mathrm{2}} \:−\:\bar {{z}}_{\mathrm{1}} }\end{vmatrix}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Arg}\:\left({z}\:−\:{z}_{\mathrm{1}} \right)\:=\:\mathrm{Arg}\:\left({z}_{\mathrm{2}} \:−\:{z}_{\mathrm{1}} \right) \\ $$

Leave a Reply

Your email address will not be published.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