Straight And Complex

Algebra Level 4

What is the reflection of the complex number 2 − i 2-i about the straight line i z = z ‾ iz=\overline z ?

Clarifications

  • z z is a complex number variable and i i represents the imaginary number, i = − 1 i=\sqrt{-1} .

  • z ‾ \overline z represents the conjugate of z z .

4 − 3 i 4-3i 1 − 2 i 1-2i 3 + 4 i 3+4i 2 + i 2+i

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Chew-Seong Cheong
Dec 30, 2015

Let z = a + b i z = a + bi

i z = z ‾ i ( a + b i ) = a − b i a i − b = a − b i ⇒ a + b − ( a + b ) i = 0 x − x i = 0 \begin{aligned} iz & = \overline{z} \\ i(a+bi) & = a-bi \\ ai - b & = a - bi \\ \Rightarrow a + b - (a+b)i & = 0 \\ x - xi & = 0 \end{aligned}

We note that i z = z ‾ iz = \overline{z} is a straight line through the origin with a gradient of − π 4 -\frac{\pi}{4} . From the figure below we find that the reflection of 2 − i 2-i about it is 1 − 2 i \boxed{1-2i} .

9 Helpful 0 Interesting 0 Brilliant 0 Confused

Equation of the mirror line should be x-xi= 0 but its x+xi = 0 in the diagram.

Mudit Jha - 5 years, 3 months ago

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Thanks, I have changed it.

Chew-Seong Cheong - 5 years, 3 months ago

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