document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210364>Question 1210364</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let z1 and z2 be two complex numbers such that |z1| = 5 and z1/z2 + z2/z1 = 0.  Find |z1 - z2|^2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852074 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852074\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve the problem:\r<BR>\n" );
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document.write( "Let $z_1$ and $z_2$ be two complex numbers.<BR>\n" );
document.write( "We are given two conditions:<BR>\n" );
document.write( "1.  $|z_1| = 5$<BR>\n" );
document.write( "2.  $\frac{z_1}{z_2} + \frac{z_2}{z_1} = 0$\r<BR>\n" );
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document.write( "From the second condition, we can multiply by $z_1 z_2$ (assuming $z_1 \neq 0$ and $z_2 \neq 0$, which must be true since $|z_1| = 5$ and if $z_2=0$ the expression is undefined):<BR>\n" );
document.write( "$z_1^2 + z_2^2 = 0$<BR>\n" );
document.write( "$z_1^2 = -z_2^2$\r<BR>\n" );
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document.write( "Now, let's take the modulus of both sides:<BR>\n" );
document.write( "$|z_1^2| = |-z_2^2|$<BR>\n" );
document.write( "$|z_1|^2 = |-1| |z_2|^2$<BR>\n" );
document.write( "$|z_1|^2 = 1 \cdot |z_2|^2$<BR>\n" );
document.write( "$|z_1|^2 = |z_2|^2$\r<BR>\n" );
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document.write( "Since $|z_1| = 5$, we have:<BR>\n" );
document.write( "$5^2 = |z_2|^2$<BR>\n" );
document.write( "$25 = |z_2|^2$<BR>\n" );
document.write( "So, $|z_2| = 5$.\r<BR>\n" );
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document.write( "Now we need to find $|z_1 - z_2|^2$.<BR>\n" );
document.write( "We know that for any complex number $z$, $|z|^2 = z \bar{z}$.<BR>\n" );
document.write( "So, $|z_1 - z_2|^2 = (z_1 - z_2)(\overline{z_1 - z_2})$<BR>\n" );
document.write( "$|z_1 - z_2|^2 = (z_1 - z_2)(\bar{z_1} - \bar{z_2})$<BR>\n" );
document.write( "$|z_1 - z_2|^2 = z_1 \bar{z_1} - z_1 \bar{z_2} - z_2 \bar{z_1} + z_2 \bar{z_2}$<BR>\n" );
document.write( "$|z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - (z_1 \bar{z_2} + z_2 \bar{z_1})$\r<BR>\n" );
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document.write( "We already know $|z_1|^2 = 25$ and $|z_2|^2 = 25$. So:<BR>\n" );
document.write( "$|z_1 - z_2|^2 = 25 + 25 - (z_1 \bar{z_2} + z_2 \bar{z_1})$<BR>\n" );
document.write( "$|z_1 - z_2|^2 = 50 - (z_1 \bar{z_2} + z_2 \bar{z_1})$\r<BR>\n" );
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document.write( "Let's go back to the relation $z_1^2 = -z_2^2$.<BR>\n" );
document.write( "We can write $z_1 = i z_2$ or $z_1 = -i z_2$.<BR>\n" );
document.write( "Case 1: $z_1 = i z_2$<BR>\n" );
document.write( "Then $\bar{z_1} = -i \bar{z_2}$.<BR>\n" );
document.write( "Substitute these into $z_1 \bar{z_2} + z_2 \bar{z_1}$:<BR>\n" );
document.write( "$z_1 \bar{z_2} + z_2 \bar{z_1} = (i z_2) \bar{z_2} + z_2 (-i \bar{z_2})$<BR>\n" );
document.write( "$= i |z_2|^2 - i |z_2|^2$<BR>\n" );
document.write( "$= 0$\r<BR>\n" );
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document.write( "Case 2: $z_1 = -i z_2$<BR>\n" );
document.write( "Then $\bar{z_1} = i \bar{z_2}$.<BR>\n" );
document.write( "Substitute these into $z_1 \bar{z_2} + z_2 \bar{z_1}$:<BR>\n" );
document.write( "$z_1 \bar{z_2} + z_2 \bar{z_1} = (-i z_2) \bar{z_2} + z_2 (i \bar{z_2})$<BR>\n" );
document.write( "$= -i |z_2|^2 + i |z_2|^2$<BR>\n" );
document.write( "$= 0$\r<BR>\n" );
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document.write( "In both cases, $z_1 \bar{z_2} + z_2 \bar{z_1} = 0$.\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "$|z_1 - z_2|^2 = 50 - 0$<BR>\n" );
document.write( "$|z_1 - z_2|^2 = 50$\r<BR>\n" );
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document.write( "The final answer is $\boxed{50}$. </SPAN>\n" );
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