Two complex numbes equotations!Heeeeelp!!?
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!!
Favorite Answer
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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