Question

Cho các số phức \(z\) thỏa \(\left|z\right|=\left|\overline{z}-1+2i\right|\). Giá trị nhỏ nhất của \(\left|\left(1+2i\right)z+11+2i\right|\) là

A. \(\dfrac{\sqrt{5}}{2}\)          B. \(\sqrt{\dfrac{5}{2}}\)           C. \(\dfrac{2}{5}\)          D. \(\dfrac{5}{2}\)

Mình cần bài giải ạ, mình cảm ơn nhiều♥

Answer 1

Gọi \(M\left(x;y\right)\) biểu diễn số phức \(z=x+yi\) \(\left(x,y\in R\right)\).

* \(\left|z\right|=\left|\overline{z}-1+2i\right|\) \(\Leftrightarrow\left|x+yi\right|=\left|x-yi-1+2i\right|\) \(\Leftrightarrow\sqrt{x^2+y^2}=\sqrt{\left(x-1\right)^2+\left(2-y\right)^2}\) \(\Leftrightarrow x^2+y^2=x^2-2x+1+4-4y+y^2\) \(\Leftrightarrow2x+4y-5=0\) \(\left(\Delta\right)\)

* \(\left|\left(1+2i\right)z+11+2i\right|\) \(=\left|z+2iz+11+2i\right|\)\(=\left|x+yi+2xi-2y+11+2i\right|\)\(=\sqrt{\left(x-2y+11\right)^2+\left(y+2x+2\right)^2}\)\(=\sqrt{\left(x+2y\right)^2+22\left(x-2y\right)+121+\left(y+2x\right)^2+4\left(y+2x\right)+4}\)\(=\sqrt{x^2-4xy+4y^2+22x-44y+121+y^2+4xy+4x^2+4y+8x+4}\)

\(=\sqrt{5x^2+5y^2+30x-40y+125}\)\(=\sqrt{x^2+y^2+6x-8y+25}\) \(=\sqrt{\left(x+3\right)^2+\left(y-4\right)^2}\) \(=MA\) với \(A\left(-3;4\right)\)

\(\rightarrow\left|\left(1+2i\right)z+11+2i\right|min=d\left(A,\left(\Delta\right)\right)\) \(=\dfrac{\left|\left(-3\right).2+4.4-5\right|}{\sqrt{2^2+4^2}}=\dfrac{\sqrt{5}}{2}\)