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Difference between revisions of "2012 AIME II Problems/Problem 8"

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== Solution ==
 
== Solution ==
 
Multiplying the two equations together gives us <cmath>zw + 32i - \frac{240}{zw} = -30 + 46i</cmath> and multiplying by <math>zw</math> then gives us a quadratic in <math>zw</math>: <cmath>(zw)^2 + (30-14i)zw - 240 =0.</cmath> Using the quadratic formula, we find the two possible values of <math>zw</math> to be <math>7i-15 \pm \sqrt{(15-7i)^2 + 240}</math> = <math>6+2i,</math> <math>12i - 36.</math> The smallest possible value of <math>\vert zw\vert^2</math> is then obviously <math>6^2 + 2^2 = \boxed{040}</math>.
 
Multiplying the two equations together gives us <cmath>zw + 32i - \frac{240}{zw} = -30 + 46i</cmath> and multiplying by <math>zw</math> then gives us a quadratic in <math>zw</math>: <cmath>(zw)^2 + (30-14i)zw - 240 =0.</cmath> Using the quadratic formula, we find the two possible values of <math>zw</math> to be <math>7i-15 \pm \sqrt{(15-7i)^2 + 240}</math> = <math>6+2i,</math> <math>12i - 36.</math> The smallest possible value of <math>\vert zw\vert^2</math> is then obviously <math>6^2 + 2^2 = \boxed{040}</math>.
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===Note===
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A key thing to note here is that <math>|zw|^2=|z|^2\cdot|w|^2,</math> which can be proved as follows:
 +
 +
Proof: Using the values for <math>z</math> and <math>w</math> that we used above, we get <math>|zw|^2=|(ac-bd)+i(bc+ad)|^2=(ac-bd)^2+(bc+ad)^2=a^2c^2+b^2d^2+b^2c^2+a^2d^2-2abcd+2abc=a^2c^2+b^2d^2+b^2c^2+a^2d^2</math>.
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Also, <math>|z|^2=a^2+b^2</math> and <math>|w|^2=c^2+d^2</math>. therefore <math>|z|^2\cdot|w|^2=(a^2+b^2)(c^2+d^2)=a^2c^2+a^2d^2+b^2c^2+b^2d^2</math> and our proof is complete <math>\Box</math>.
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Now, also note that we found <math>\sqrt{416-210i}</math> by letting <math>416-210i=(a-bi)^2</math> and solving for <math>a</math> and <math>b</math> by considering real and imaginary parts. Then, we substitute that into <math>a-bi</math> which is the value of <math>\sqrt{416-210i}</math> and continue from there.
  
 
== See Also ==
 
== See Also ==
 
{{AIME box|year=2012|n=II|num-b=7|num-a=9}}
 
{{AIME box|year=2012|n=II|num-b=7|num-a=9}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:52, 14 January 2024

Problem 8

The complex numbers $z$ and $w$ satisfy the system \[z + \frac{20i}w = 5+i\] \[w+\frac{12i}z = -4+10i\] Find the smallest possible value of $\vert zw\vert^2$.

Solution

Multiplying the two equations together gives us \[zw + 32i - \frac{240}{zw} = -30 + 46i\] and multiplying by $zw$ then gives us a quadratic in $zw$: \[(zw)^2 + (30-14i)zw - 240 =0.\] Using the quadratic formula, we find the two possible values of $zw$ to be $7i-15 \pm \sqrt{(15-7i)^2 + 240}$ = $6+2i,$ $12i - 36.$ The smallest possible value of $\vert zw\vert^2$ is then obviously $6^2 + 2^2 = \boxed{040}$.

Note

A key thing to note here is that $|zw|^2=|z|^2\cdot|w|^2,$ which can be proved as follows:

Proof: Using the values for $z$ and $w$ that we used above, we get $|zw|^2=|(ac-bd)+i(bc+ad)|^2=(ac-bd)^2+(bc+ad)^2=a^2c^2+b^2d^2+b^2c^2+a^2d^2-2abcd+2abc=a^2c^2+b^2d^2+b^2c^2+a^2d^2$.

Also, $|z|^2=a^2+b^2$ and $|w|^2=c^2+d^2$. therefore $|z|^2\cdot|w|^2=(a^2+b^2)(c^2+d^2)=a^2c^2+a^2d^2+b^2c^2+b^2d^2$ and our proof is complete $\Box$.

Now, also note that we found $\sqrt{416-210i}$ by letting $416-210i=(a-bi)^2$ and solving for $a$ and $b$ by considering real and imaginary parts. Then, we substitute that into $a-bi$ which is the value of $\sqrt{416-210i}$ and continue from there.

See Also

2012 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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