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# Complex Numbers

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Let $$z_1$$and $$z_2$$ be complex numbers such that $$\frac{z_2}{z_1}$$ is pure imaginary and $$2z_1 \neq 7z_2$$. Compute $$\left| \frac{2z_1 + 7z_2}{2z_1 - 7z_2} \right|.$$

Jul 7, 2022

#1
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Let z_1 = a + bi and z_2 = c + di.  Then z_2/z_1 = pure imaginary gives you d/a = -c/b.  Let d/a = k and c/b = -k.

Then d = ka and c = -bk.  A little bit of algebra then gives you

$$\left| \frac{2z_1 + 7z_2}{2z_1 - 7z_2} \right| = \frac{2}{7}$$

Jul 7, 2022
#2
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I don't think this is right?

Guest Jul 7, 2022
#3
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$$Let \qquad ki=\frac{z_2}{z_1} \qquad k\in Z\\ \displaystyle \left| \frac{2z_1+7z_2}{2z_1-7z_2}\right|\\ =\displaystyle \left| \frac{\frac{2z_1+7z_2}{z_1}}{\frac{2z_1-7z_2}{z_1}}\right|\\ =\displaystyle \left| \frac{2+7ki}{2-7ki}\right|\\ =\displaystyle \left| \frac{(2+7ki)^2}{4+49k^2}\right|\\ =\displaystyle \left| \frac{(4-49k^2)+28ki}{4+49k^2}\right|\\ =\displaystyle \frac{\sqrt{(4-49k^2)^2+(28k)^2}}{4+49k^2}\\ =\displaystyle \frac{\sqrt{ 16+2401k^4-392k^2 +784k^2}}{4+49k^2}\\ =\displaystyle \frac{\sqrt{ 2401k^4+392k^2+16 }}{4+49k^2}\\ =\displaystyle \frac{ 49k^2+4 }{4+49k^2}\\ =1$$

You need to check that.

\displaystyle \left|  \frac{2z_1+7z_2}{2z_1-7z_2}\right|\\
=\displaystyle \left|  \frac{\frac{2z_1+7z_2}{z_1}}{\frac{2z_1-7z_2}{z_1}}\right|\\
=\displaystyle \left|  \frac{2+7ki}{2-7ki}\right|\\
=\displaystyle \left|  \frac{(2+7ki)^2}{4+49k^2}\right|\\
=\displaystyle \left|  \frac{(4-49k^2)+28ki}{4+49k^2}\right|\\
=\displaystyle   \frac{\sqrt{(4-49k^2)^2+(28k)^2}}{4+49k^2}\\
=\displaystyle   \frac{\sqrt{   16+2401k^4-392k^2      +784k^2}}{4+49k^2}\\
=\displaystyle   \frac{\sqrt{   2401k^4+392k^2+16 }}{4+49k^2}\\
=\displaystyle   \frac{  49k^2+4 }{4+49k^2}\\
=1

Jul 8, 2022