Question

Cho \(\left|z-3+4i\right|=\left|z\right|\). Tìm z để \(\left|z\right|_{min}\) và \(\left|z\right|_{max}\)

Answer 1

\(z=x+yi\Rightarrow\sqrt{\left(x-3\right)^2+\left(y+4\right)^2}=\sqrt{x^2+y^2}\)

\(\Rightarrow6x-8y-25=0\)

\(\Rightarrow y=\dfrac{6x-25}{8}\)

\(\Rightarrow\left|z\right|=\sqrt{x^2+\left(\dfrac{6x-25}{8}\right)^2}=\dfrac{5}{8}\sqrt{\left(2x-3\right)^2+16}\ge\dfrac{5}{2}\)

Dấu "=" xảy ra khi \(x=\dfrac{3}{2};y=-2\Rightarrow z=\dfrac{3}{2}-2i\)

Không tồn tại \(\left|z\right|_{max}\)