1.

The minimum value of |z_(1)-z_(2)| as z_(1) and z_(2) vary over the curves |sqrt(3)(1-2z)+2i|=2sqrt(7) and |sqrt(3)(-1-z)-2i|=|sqrt(3)(9-z)+18i| respectively is

Answer»

`(7sqrt(7))/(2sqrt(3))`
`(5sqrt(7))/(2sqrt(3))`
`(14sqrt(7))/(sqrt(3))`
`(7sqrt(7))/(5sqrt(3))`

Solution :We have,
`|sqrt(3)(1-2z)+2i|=2sqrt(7)`
`|sqrt(3)(1-2z)+2i|=2sqrt(7)`
`rArr |-2sqrt(3z)+(sqrt(3)+2i)|=2sqrt(7)`
`rArr |Z-(1/2+1/sqrt(3)i)|=sqrt(7/3)`
Clearly, it represents a CIRCLE haivng center at `(1//2,1//sqrt(3))` and radius `r_(1)=sqrt(7/3)`. it is given that `z_(1)` LIES on (i)

The equation of another curve is
`|sqrt(3)(-1-z)-2i|=|sqrt(3)(9-z)+18i|`
or `|-1-z-2/sqrt(3)i|=|9-z+6sqrt(3)i|`
or `,|z+1+2/sqrt(3)i|=|z-9-6sqrt(3)i|`
or `,|z-(-1-2/sqrt(3)i)|=|z-(9+6sqrt(3)i)|`
This, represents perpendicular bisector of the line segement joining point `A(-1,-2/sqrt(3))` and `B(9,6sqrt(3))`. The coordinates of the mid-point C of AB are `(4,8//sqrt(3))`. Clearly, A,B and the center of the circle are collinear.


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