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Algebra Level 4

Let z 1 z_1 and z 2 z_2 be two given complex numbers such that z 1 z 2 + z 2 z 1 = 1 \dfrac{z_1}{z_2} + \dfrac{z_2}{z_1} = 1 and z 1 = 3 |z_1| = 3 . Compute z 1 z 2 2 |z_1 - z_2|^2 .


The answer is 9.

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2 solutions

Sujoy Roy
Jan 10, 2016

Putting z 1 z 2 = t \frac{z_1}{z_2}=t , we get t + 1 t = 1 t+\frac{1}{t}=1 or, t 2 t + 1 = 0 t^2-t+1=0 or, t = 1 + 3 i 2 t=\frac{1+\sqrt{3}i}{2} or, t = 1 |t|=1 or, z 1 = z 2 = 3 |z_1|=|z_2|=3 .

Now z 1 z 2 2 = ( z 1 z 2 ) ( z 1 ˉ z 2 ˉ ) |z_1-z_2|^2= (z_1-z_2)(\bar{z_1}-\bar{z_2})

= z 1 z 1 ˉ + z 2 z 2 ˉ ( z 1 z 2 ˉ + z 2 z 1 ˉ ) = z_1\bar{z_1}+z_2\bar{z_2}-(z_1\bar{z_2}+z_2\bar{z_1})

= 9 + 9 ( t z 2 z 2 ˉ + z 1 z 1 ˉ t ) =9+9-(tz_2\bar{z_2}+\frac{z_1\bar{z_1}}{t})

= 18 9 ( t + 1 t ) = 18 9 = 9 =18-9(t+\frac{1}{t})=18-9=\boxed{9} .

but z 1 z 1 ˉ = z 1 2 { z }_{ 1 }\bar { { z }_{ 1 } } ={ \left| { z }_{ 1 } \right| }^{ 2 }

Tanishq Varshney - 5 years, 5 months ago

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You are absolutely correct Tanishq !

Bob Kadylo - 5 years, 4 months ago

The answer should be 9 \boxed 9 because there is an exponent 2 outside the z 1 z 2 2 |z_1-z_2|^2

Bob Kadylo - 5 years, 4 months ago
Zk Lin
Feb 11, 2016

From z 1 z 2 + z 2 z 1 = 1 \frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{1}}=1 , we have

z 1 2 + z 2 2 z 1 z 2 = 1 \frac{{z_{1}}^{2}+{z_{2}}^{2}}{z_{1}z_{2}}=1

z 1 2 + z 2 2 = z 1 z 2 {z_{1}}^{2}+{z_{2}}^{2}=z_{1}z_{2}

( z 1 z 2 ) 2 = z 1 z 2 (z_{1}-z_{2})^{2}=-z_{1}z_{2}

{|z_{1}-z_{2}|}^{2}=|z_{1}||z_{2}| \ \tag{1}

From z 1 2 + z 2 2 = z 1 z 2 {z_{1}}^{2}+{z_{2}}^{2}=z_{1}z_{2} , we have

z 1 2 = z 2 z 1 z 2 2 {z_{1}}^{2}=z_{2}z_{1}-{z_{2}}^{2}

z 1 2 = z 2 ( z 1 z 2 ) {z_{1}}^{2}=z_{2}(z_{1}-z_{2})

z 1 2 z 2 = z 1 z 2 \frac{{|z_{1}|}^{2}}{|z_{2}|}=|z_{1}-z_{2}|

\frac{{|z_{1}|}^{4}}{{|z_{2}|}^{2}}={|z_{1}-z_{2}|}^{2} \ \tag{2}

( 1 ) = ( 2 ) (1)=(2)

z 1 4 z 2 2 = z 1 z 2 \frac{{|z_{1}|}^{4}}{{|z_{2}|}^{2}}=|z_{1}||z_{2}|

z 1 3 = z 2 3 {|z_{1}|}^{3}={|z_{2}|}^{3}

Therefore,

z 1 = z 2 = 3 |z_{1}|=|z_{2}|=3

z 1 z 2 2 = z 1 z 2 = 9 {|z_{1}-z_{2}|}^{2}=|z_{1}||z_{2}|=\boxed{9}

Moderator note:

The explanation is correct, but it seems hard to motivate such an approach.

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