Verified Solution • 2025–26 Session moderate QJEEAdv10Ma05c5: JEE Advanced 2010 Mathematics - Class 11 JEEAdv10Ma05c5 Solution: Let and be two distinct complex numbers let for some real… Let z1z_1z1 and z2z_2z2 be two distinct complex numbers let z=(1−t)z1+tz2z=(1-t) z_1+t z_2z=(1−t)z1+tz2 for some real number t with 0 < t < 1 . If Arg(w)\operatorname{Arg}(w)Arg(w) denotes the principal argument of a nonzero complex number www , then : Multiple Choice Options A ∣z−z1∣+∣z−z2∣=∣z1−z2∣\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right|∣z−z1∣+∣z−z2∣=∣z1−z2∣ B Arg(z−z1)=Arg(z−z2)\operatorname{Arg}\left(z-z_1\right)=\operatorname{Arg}\left(z-z_2\right)Arg(z−z1)=Arg(z−z2) C ∣z−z1zˉ−zˉ1z2−z1zˉ2−zˉ1∣=0\left|\begin{array}{cc}z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1\end{array}\right|=0z−z1z2−z1zˉ−zˉ1zˉ2−zˉ1=0 D Arg(z−z1)=Arg(z2−z1)\operatorname{Arg}\left(z-z_1\right)=\operatorname{Arg}\left(z_2-z_1\right)Arg(z−z1)=Arg(z2−z1) Step-by-Step Verified Solution amp; Given, z=(1−t)z1+tz2amp;⇒z−z1z2−z1=tamp;⇒arg(z−z1z2−z1)=0.......(i)\begin{aligned} & \text { Given, } z=(1-t) z_1+t z_2 \\\\ & \Rightarrow \frac{z-z_1}{z_2-z_1}=t \\\\ & \Rightarrow \arg \left(\frac{z-z_1}{z_2-z_1}\right) = 0 .......(i) \end{aligned}amp; Given, z=(1−t)z1+tz2amp;⇒z2−z1z−z1=tamp;⇒arg(z2−z1z−z1)=0.......(i) ⇒arg(z−z1)amp;=arg(z2−z1)z−z1z2−z1amp;=zˉ−zˉ1zˉ2−zˉ1∣z−z1amp;zˉ−zˉ1z2−z1amp;zˉ2−zˉ1∣amp;=0\begin{aligned} \Rightarrow \arg \left(z-z_1\right) & =\arg \left(z_2-z_1\right) \\\\ \frac{z-z_1}{z_2-z_1} & =\frac{\bar{z}-\bar{z}_1}{\bar{z}_2-\bar{z}_1} \\\\ \left|\begin{array}{cc}z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1\end{array}\right| & =0\end{aligned}⇒arg(z−z1)z2−z1z−z1z−z1z2−z1amp;zˉ−zˉ1amp;zˉ2−zˉ1amp;=arg(z2−z1)amp;=zˉ2−zˉ1zˉ−zˉ1amp;=0 AP+PB=AB⇒∣z−z1∣+∣z−z2∣=∣z1−z2∣\begin{gathered}\mathrm{AP}+\mathrm{PB}=\mathrm{AB} \\\\ \Rightarrow\left|z-z_1\right|+\left|z-z_2\right|=\left|z_1-z_2\right|\end{gathered}AP+PB=AB⇒∣z−z1∣+∣z−z2∣=∣z1−z2∣ Related Learning Resources & Exam Preparation JEE Advanced PYQ Question Bank Browse year-wise authentic previous years' questions with solutions Authentic CBT Mock Tests Simulate official exam environment with live timer and percentile scoring
