Difference between revisions of "1988 AHSME Problems/Problem 21"

(Created page with "==Problem== The complex number <math>z</math> satisfies <math>z + |z| = 2 + 8i</math>. What is <math>|z|^{2}</math>? Note: if <math>z = a + bi</math>, then <math>|z| = \sqrt{a^{...")
 
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==Solution==
 
==Solution==
  
 
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Let the complex number <math>z</math> equal <math>a+bi</math>. Then the preceding equation can be expressed as <cmath>a+bi+\sqrt{a^2+b^2} = 2+8i</cmath> Because <math>a</math> and <math>b</math> must both be real numbers, we immediately have that <math>bi = 8i</math>, giving <math>b = 8</math>. Plugging this in back to our equation gives us <math>a+\sqrt{a^2+64} = 2</math>. 
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Rearranging this into <math>2-a = \sqrt{a^2+64}</math>, we can square each side of the equation resulting in <cmath>4-4a+a^2 = a^2+64</cmath> Further simplification will yield 
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<math>60 = -4a</math> meaning that <math>-15 = a</math>. Knowing both <math>a</math> and <math>b</math>, we can plug them in into <math>a^2+b^2</math>. Our final answer is <math>\boxed{289}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 20:58, 8 June 2016

Problem

The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.

$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$


Solution

Let the complex number $z$ equal $a+bi$. Then the preceding equation can be expressed as \[a+bi+\sqrt{a^2+b^2} = 2+8i\] Because $a$ and $b$ must both be real numbers, we immediately have that $bi = 8i$, giving $b = 8$. Plugging this in back to our equation gives us $a+\sqrt{a^2+64} = 2$. Rearranging this into $2-a = \sqrt{a^2+64}$, we can square each side of the equation resulting in \[4-4a+a^2 = a^2+64\] Further simplification will yield $60 = -4a$ meaning that $-15 = a$. Knowing both $a$ and $b$, we can plug them in into $a^2+b^2$. Our final answer is $\boxed{289}$.

See also

1988 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
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