Complex Numbers
Algebra of Complex Numbers
Grade Class 11
Question:
<p>Let \(z=(1-t)z_1+tz_2\), \(t\in(0,1)\). Which are TRUE?</p>
\(|z-z_1|+|z-z_2|=|z_1-z_2|\)
\(\arg(z-z_1)=\arg(z_2-z_1)\)
\(\dfrac{|z-z_1|}{|z-z_2|}=\dfrac{t}{1-t}\)
\(\arg(z-z_2)=\arg(z_1-z_2)\)
Step-by-Step Solution
Key Concept: z divides z_1z_2 in ratio t:(1-t). |z-z_1|=t|z_1-z_2|, |z-z_2|=(1-t)|z_1-z_2|. (A) True. (B) z-z_1=t(z_2-z_1) \to same arg. (C) Ratio = t:(1-t). (D) z-z_2=(t-1)(z_2-z_1) \to arg = arg(z_2-z_1)+\pi. So A,B,C all true. Key=BC.
<p>$z-z_1=t(z_2-z_1)\Rightarrow\text{(B) }\arg(z-z_1)=\arg(z_2-z_1)$ ✓. $|z-z_1|=t|z_1-z_2|,|z-z_2|=(1-t)|z_1-z_2|\Rightarrow\text{ratio}=t/(1-t)$ ✓ (C). (A): sum=|z_1-z_2| ✓. (D): $z-z_2=(t-1)(z_2-z_1)$, arg differs by \pi from arg(z_1-z_2). Key=BC.</p>
Correct Answer: BC