IIT-JEE 2010 · Paper 1 · Mathematics Q39

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MathematicsMultiple correct+3Question 39 of 84

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 nonzero complex number w, then

|z-z_1|+|z-z_2|=|z_1-z_2|

\operatorname{Arg}(z-z_1)=\operatorname{Arg}(z-z_2)

\begin{vmatrix}z-z_1&\bar z-\bar z_1\\z_2-z_1&\bar z_2-\bar z_1\end{vmatrix}=0

\operatorname{Arg}(z-z_1)=\operatorname{Arg}(z_2-z_1)

Answer and worked solution

Official answer: A, C, D

$z$ lies strictly between $z_1$ and $z_2$ on the segment: $z-z_1=t(z_2-z_1)$ and $z-z_2=(1-t)(z_1-z_2)$ , with $0<t<1$ .

(A) The distances along the segment add up: $|z-z_1|+|z-z_2|=|z_1-z_2|$ . True.
(B) $z-z_1$ and $z-z_2$ point in opposite directions, so their arguments differ by $\pi$ . False.
(C) The determinant is $(z-z_1)(\bar z_2-\bar z_1)-(\bar z-\bar z_1)(z_2-z_1)=t|z_2-z_1|^2-t|z_2-z_1|^2=0$ . True.
(D) $z-z_1$ is a positive multiple of $z_2-z_1$ , so they have the same argument. True.

(A), (C), (D)

Topics:Geometry in the complex plane Modulus and argument

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IIT-JEE 2010 · Paper 1, Mathematics question 39