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If |z-1| + |z + 3| le 8, then prove that...

If `|z-1| + |z + 3| le 8`, then prove that z lies on the circle.

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The correct Answer is:
`1 le |z-4|le9`

Given `|z-1| + | z + 3| le 8`. Then z lies inside or on the ellipse whose foci are (1,0) and (-3, 0) and vertices are (-5, 0) and (3,0).
Clearly the minmum and maximum values of the distance PA and PA. Thus, `1 le |z - 4| le 9`.
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