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I've been struggling with this complex numbers question for some time now. I've tried converting each of the parts to cartesian form but it keeps cancelling out. Could anyone help please. Thanks in advance.

Problem

Edit: I have found the line representing the perpendicular bisector to be Re(z)=2d^2+18. I have found that the equation representing the half line can be represented as Im(z)=Re(z)+3-12d. I'm struggling to go from here.

MathIsFun
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  • $C_1$ is the locus of points at equal distance from $0$ and $4d^2-36$, which is the perpendicular bisector of the segment between the two points. That's a vertical line in the complex plane, so this gives you $\Re(z)$. Then find the matching point on $C_2$, which is a ray parallel to the first bisector. – dxiv Jun 06 '21 at 21:06
  • That sounds good. However, I am struggling to find the equation of the perpendicular bisector. Is it just 0? – MathIsFun Jun 06 '21 at 21:15
  • The perpendicular bisector is not a point, it's a line (except in the edge case where the two endpoints coincide). You should edit your question and show what you tried and where you got stuck. – dxiv Jun 06 '21 at 21:29
  • Yes I've done that now @dxiv – MathIsFun Jun 06 '21 at 21:33
  • All that's left is to solve the system of two equations with two unknowns $\Re(z)$ and $\Im(z)$. Actually, the first equation gives you $\Re(z)$ directly. Then you will have found $z=\Re(z) + i,\Im(z)$. – dxiv Jun 06 '21 at 21:36
  • So how would I represent the final answer? Do I just sub in the Re(z) into Im(z) to get the imaginary. Then I get the two parts and represent it as Re(z)+i*Im(z)? – MathIsFun Jun 06 '21 at 21:38
  • Ok great actually. I got it. Thank you @dxiv! – MathIsFun Jun 06 '21 at 21:45
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    Glad it helped. Don't forget "the edge case where the two endpoints coincide", though. – dxiv Jun 06 '21 at 21:51

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