If the complex number z is such that |z−$$25$$|≤$$25$$ and |z−$$50$$|$$=25$$ then the minimum value of |z|$$+25$$ is $$10k$$ then k is equal to

Geometry of complex numbers

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Question

If the complex number z is such that $| z - 25 | \leq 25$ and $| z - 50 | = 25$ then the minimum value of $| z | + 25$ is 10k then k is equal to

Moderate

Solution

$| z - 25 | \leq 25$ represent interior and boundary of circle with
centre (25, 0) and radius 25. Points z satisfying $| z - 25 | \leq 25$
and $| z - 50 | = 25$ lie on arc DAC hence min ${| z |} = OA = 25$
$∴ | z | + 25 = 50 \cdot So, k = 5$

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