Question

cho số phức z thoả điều kiện  |3z+3-4i|=5 . tìm z sao cho |3z-i| đạt gtnn

Answer 1

Ta có : \(\left|3z-i\right|+\left|3-3i\right|\ge\left|3z+3-4i\right|=5\)

\(\Rightarrow\left|3z-i\right|+3\sqrt{2}\ge5\) \(\Rightarrow\left|3z-i\right|\ge5-3\sqrt{2}\)

" = " \(\Leftrightarrow3z-i=k\left(3-3i\right)\left(k>0\right)\)

\(\Leftrightarrow3z=k\left(3-3i\right)+i\)

Khi đó : \(\left|k\left(3-3i\right)+i+3-4i\right|=5\)

\(\Leftrightarrow2\left(3k+3\right)^2=25\) \(\Leftrightarrow18\left(k+1\right)^2=25\)

\(\Leftrightarrow\left[{}\begin{matrix}k=\dfrac{5\sqrt{2}}{6}-1\\k=\dfrac{5\sqrt{2}}{6}+1\end{matrix}\right.\)   ( t/m )

\(\Rightarrow z=...\)