Find the locus of complex number
$z = x + i y$ if $| z + 4 i | + | z - 4 i | = 10$

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Solution

$| z + 4! | + | z - 4! | = 10$
$\sqrt{x^{2} + (y - y)^{2}} + \sqrt{x^{2} + (y + 4)^{2}} = 10$
$\sqrt{x^{2} + (y - 4)^{2}} = 10 - \sqrt{x^{2} + (y + 4)^{2}}$
$x^{2} + y^{2} + 16 - 8 y = 100 + x^{2} + y^{2} + 16 + 8 y - 20 \sqrt{x^{2} (y + 4)^{2}}$
$4 y + 25 = 5 \sqrt{x^{2} + (y + 4)^{2}}$
$16 x^{2} + 625 + 200 y = 25 x^{2} + 25 y^{2} + 200 y + 400$
$25 x^{2} + 9 y^{2} = 225$

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