Min.value of Complex no.

[man111]

If [tex]Z[/tex] be a Complex no. such that [tex]|Z-3i|+|Z-4|=5[/tex].Then find Minimum value of [tex]|Z|[/tex]

[mkmarinov]

[tex]|Z-3i|+|Z-4| \ge |3i+4|=5[/tex] => Z lies on the line, connecting the points, defined by 3i and 4. Let us call the line a.
The modulus of Z is the distance between Z and the origin, but Z could be any point on a, so the least distance is the one between the origin and the line a. Which is an altitude in a certain right-angled triangle .
[tex]\frac{12}{5}[/tex]

[man111]

Thanks mkmarinov.