A few days ago
Joey

Two complex numbes equotations!Heeeeelp!!?

The first one:

1)solve the equotation:z^2+z*^2=zz* and then find the locus of the points of its solutions.

2)solve the equotation:z^2+4z=z*^2+4z… and then find the locus of the points of its solutions.

I really hope i made the questions clear!!

Top 1 Answers
A few days ago
Anonymous

Favorite Answer

1.

Putting z = x + iy, z* = x – iy:

(x + iy)^2 + (x – iy)^2 = (x + iy)(x – iy)

x^2 – y^2 + 2ixy + x^2 – y^2 – 2ixy = x^2 + y^2

2x^2 – 2y^2 = x^2 + y^2

x^2 – 3y^2 = 0

Hence:

(x – y sqrt(3))(x + y sqrt(3)) = 0

y = +/- x / sqrt(3)

y = +/- x sqrt(3) / 3.

The locus is two straight lines of gradient +/- sqrt(3) / 3 both passing through the origin.

2.

z^2+4z=z*^2+4z*

(x + iy)^2 + 4x + 4iy = (x – iy)^2 + 4x – 4iy

x^2 – y^2 + 2ixy + 4x + 4iy = x^2 – y^2 – 2ixy + 4x – 4iy

6iy = -6iy

y = -y

2y = 0

y = 0.

The locus is the x-axis.

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