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2021 AMC 12B Problems/Problem 18

Revision as of 19:50, 11 February 2021 by Lopkiloinm (talk | contribs) (Created page with "==Problem 18== Let <math>z</math> be a complex number satisfying <math>12|z|^2=2|z+2|^2+|z^2+1|^2+31.</math> What is the value if <math>z+\frac 6z?</math> <math>\textbf{(A) }...")
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Problem 18

Let $z$ be a complex number satisfying $12|z|^2=2|z+2|^2+|z^2+1|^2+31.$ What is the value if $z+\frac 6z?$

$\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4$

The answer being in the form $z+\frac 6z$ means that there are two solutions, some complex number and its complex conjugate. \[a+bi = \frac{6}{a-bi}\] \[a^2+b^2=6\]. We should then be able to test out some ordered pairs of (a, b). After testing it out, we get the ordered pairs of $(-1, \sqrt{5})$ and its conjugate $(-1, -\sqrt{5})$. Plugging this into answer format gives us $\textbf{(A) }-2$

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