Let z₁ and z₂ be two distinct complex numbers and let z=(1-t) z₁+t z₂ for some real number t with 0 < t < 1 . If (w) denotes the principal argument of a non-zero complex number w , then

Step-by-step solution

Given, z= (1-t) z₁+t z₂ (1-t)+t Clearly, z divides z₁ and z₂ in the ratio of t:(1-t), 0 < t < 1 aligned & A P+B P=A B & ie, |z-z₁ |+ |z-z₂ |= |z₁-z₂ | & Option (a) is true. & and aligned (z-z₁ ) & = (z₂-z ) & = (z₂-z₁ ) aligned aligned (b) is false and (d) is true. Also, (z-z₁ )= (z₂-z₁ ) aligned & ( z-z₁ z₂-z₁ )=0 & z-z₁ z₂-z₁ is purely real. & z-z₁ z₂-z₁ = z - z ₁ z ₂- z ₁ & or | array cc z-z₁ & z - z ₁ z₂-z₁ & z ₂- z ₁ array |=0 & aligned Option (c) is correct. Hence, (a, c, d) is the correct option.