Question

[Hoang Nguyen Dat]

Cho số phức z thỏa mãn\(\left|z-\overline{z}+2i\right|=\left|\dfrac{3}{2}z+\dfrac{1}{2}\overline{z}\right|\)^{. Tìm giá trị nhỏ nhất của \(P=\left|z-3\right|\)}

Answer 1

[Mysterious Person]

đặc \(z=a+bi\) với \(\left(a;b\in R;i^2=-1\right)\)

ta có : \(\left|z-\overline{z}+2i\right|=\left|\dfrac{3}{2}z+\dfrac{1}{2}\overline{z}\right|\)

\(\Leftrightarrow\left|a+bi-a+bi+2i\right|=\left|\dfrac{3}{2}a+\dfrac{3}{2}bi+\dfrac{1}{2}a-\dfrac{1}{2}bi\right|\)

\(\Leftrightarrow\sqrt{\left(2b+2\right)^2}=\sqrt{\left(2a+b\right)^2}\) \(\Leftrightarrow2b+2=2a+b\Leftrightarrow a=\dfrac{b}{2}+1\)

ta có : \(P=\left|z-3\right|=\left|a+bi-3\right|=\sqrt{\left(a-3\right)^2+b^2}\)

\(\Leftrightarrow P=\sqrt{\left(\dfrac{b}{2}+1-3\right)^2+b^2}=\sqrt{\left(\dfrac{b}{2}-2\right)^2+b^2}\)

\(\Leftrightarrow P=\sqrt{\dfrac{5b^2}{4}-2b+4}\ge\sqrt{4-\dfrac{\left(-2\right)^2}{4.\dfrac{5}{4}}}=\dfrac{4\sqrt{5}}{5}\)

dấu "=" xảy ra khi \(b=\dfrac{2}{2.\dfrac{5}{4}}=\dfrac{4}{5}\) và \(a=\dfrac{7}{5}\) \(\Leftrightarrow z=\dfrac{7}{5}+\dfrac{4}{5}i\)