[JEE Advanced 2003] If $|z_1|<1<|z_2|$, prove that $\left|\dfrac{1-z_1\bar z_2}{z_1-z_2}\right|<1$.
If $|z_1|<1<|z_2|$, prove that $\left|\dfrac{1-z_1\bar z_2}{z_1-z_2}\right|<1$.
1 Answer
VAVidaara Admin
✓ Vidaara Team
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· 1mo ago
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Answer: Proved.
$|z_1-z_2|^2-|1-z_1\bar z_2|^2=(|z_1|^2-1)(1-|z_2|^2)$. With $|z_1|<1<|z_2|$ both factors are negative, so the product is positive: $|z_1-z_2|>|1-z_1\bar z_2|$, giving the ratio $<1$.
JEE Advanced 2003 · Complex Numbers — verified solution by the Vidaara Team.
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