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To find the equation of the ellipse with foci $\pm3i$ that goes through the point $8-3i$.

The standard equation of ellipse given foci is $|z-f|+|z+f|=2 \mu |f|$, where $\pm f$ is foci and $z$ is a point on the ellipse.

So, $|8-3i-3i|+|8-3i+3i|=2 \mu |3i| \implies 18=2 \times3 \mu \implies\mu=3.$

Thus the equation of ellipse is: $|z-3i|+|z+3i|=6|3i| = 18.$

Is the solution correct?

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More simply: the sum of the distances from a point of the ellipse to the foci should be constant: since $$ |8-3i-3i|+|8-3i+3i|=18 $$ the equation is $$ |z-3i|+|z+3i|=18 $$ No need to compute $\mu$.

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