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If $|z-5\sqrt3-5i|=5$, then find $a+b+c+d$ such that $\left|\frac1z-\frac1{a\sqrt{b}}+\frac{i}{c}\right|=\frac1d$.

I try to make the second equation look like the first one. Multiplying both sides by $acz\sqrt{b}$, we get $$|ac\sqrt{b} - cz + az\sqrt{b}i| = \frac{acz\sqrt{b}}{d}$$ Now we have $z$ in the numerator, which is good. Factoring $z$ out from the second and third terms in the "absolute value" signs, we get $$|z(-c + az\sqrt{b}i) + ac\sqrt{b}| = \frac{acz\sqrt{b}}{d}$$ Dividing both sides by $(-c + az\sqrt{b}i)$, we get $$\left|z + \frac{ac\sqrt{b}}{-c + az\sqrt{b}i}\right| = \left|z - \frac{ac\sqrt{b}}{c - az\sqrt{b}i}\right| = \frac{acz\sqrt{b}}{d(-c+az\sqrt{b}i)}$$ Looking at the 2 rightmost expressions, it resembles $|z - (5\sqrt{3} + 5i)| = 5$. We get $$\frac{ac\sqrt{b}}{c - az\sqrt{b}i}=5\sqrt{3}+5i$$ and$$\frac{ac\sqrt{b} \cdot z}{(-c+az\sqrt{b}i) \cdot d} = 5$$ Dividing the equations gives $\frac{d}{z} = \sqrt{3} + i$.

But then I don't know how to combine $\frac{d}{z}=\sqrt{3}+i$ and $\frac{ac\sqrt{b}}{c - az\sqrt{b}i}=5\sqrt{3}+5i$ to find $a + b + c + d$.

Note: Please do not provide any answers using material beyond Algebra 2.

py_math
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    You did not multiply both sides by $acz\sqrt{b}$. You multiplied the left side by $|acz\sqrt{b}|$ and the right side by $acz\sqrt{b}$. I'd start out using that $|w|^2 = w \bar{w}$. – aschepler May 03 '23 at 22:14
  • This is one of those cases where it's probably easier to solve the general problem, first: given $|z-u|=v$, find $w,t$ such that $\left|\frac{1}{z}-w\right|=t$. – dxiv May 03 '23 at 23:58
  • The given equation is that of a circle. It can be rewritten as a quadratic equation in variables $(x,y),$ or in complex numbers with $z$ and $\overline{z}.$ This text can really help, but there are several typos, be careful. – user376343 May 04 '23 at 21:28
  • I don't know what matrices are. – py_math May 04 '23 at 21:55
  • Does this answer your question? Magnitude of Complex Number And Other Equation – dxiv May 05 '23 at 23:11
  • No, I don't understand the inversion part of the answer and why it works. – py_math May 06 '23 at 01:15
  • @py_math You don't really need to know about inversion in advance. The problem tells you that the transformation $z \mapsto \frac{1}{z}$ maps the original circle to another circle. The image circle is defined by $3$ points, so you can choose three convenient ones, then determine the center and radius. See also transforming circles into circles. – dxiv May 06 '23 at 07:22

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