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Question

Mathematics Question on Complex Numbers and Quadratic Equations

The value of z2+z32+zi2|z|^2 + |z - 3|^2 + |z - i|^2 is minimum when zz equals.

A

223i2 \,\frac {2} {3} i

B

45+3i45 + 3i

C

1+i31+ \frac {i} {3}

D

1i31 \,\frac {i} {3}

Answer

1+i31+ \frac {i} {3}

Explanation

Solution

z2+z32+z12\left|z\right|^{2}+\left|z-3\right|^{2}+\left|z-1\right|^{2} =x2+y2+(x3)2+y2+x2(y1)2=x^{2}+y^{2}+\left(x-3\right)^{2}+y^{2}+x^{2}\left(y-1\right)^{2} =3x2+3y26x2y+10=3x^{2}+3y^{2}-6x-2y+10 =3(x1)2+3(y22y3)+103=3\left(x-1\right)^{2}+3\left(y^{2}-\frac{2y}{3}\right)+10-3 =3(x1)2+3(y223)2+73=3\left(x-1\right)^{2}+3\left(y^{2}-\frac{2}{3}\right)^{2}+7-3 =3z(1+i3)2+203=3\left|z-\left(1+\frac{i}{3}\right)\right|^{2}+\frac{20}{3} This is minimum if z=1+i3z=1+\frac{i}{3}