Question

# If $$\displaystyle \left | z-1 \right |+\left | z+3 \right |\leq 8$$,then range of $$\left | z-4 \right |$$ is

A
(0,5)
B
(7,0)
C
(1,9)
D
(5,0)

Solution

## The correct option is C $$\left ( 1,9 \right )$$Given:$$\left| z+3 \right| +\left| z-1 \right| \le 8$$z lies on and inner region of the ellipse $$\left| z+3 \right| +\left| z-1 \right| =8$$ From the figure:Equation of ellipse is $$SP+ S'P =2a$$ where, S & S' are focii and $$2a$$ is major axis.$$\left| z+3 \right| +\left| z-1 \right| =8$$Comparing above eq with $$SP+S'P=2a$$ we get, $$2a=8$$$$\Rightarrow a=4$$$$min\left| z-4 \right|$$=distance between point $$P(4,0)$$ & point $$A(3,0)=d(PA)$$$$min\left| z-4 \right| =d(PA)=1$$$$max\left| z-4 \right|=$$distance between point $$P(4,0)$$ & point $$B(-5,0)=d(PB)$$$$max\left| z-4 \right|=d(PB)=9$$$$min\left| z-4 \right| \le \left| z-4 \right| \le max\left| z-4 \right|$$$$\Rightarrow 1\le \left| z-4 \right| \le 9$$$$\Rightarrow \left| z-4 \right| \in \left[ 1,9 \right]$$Hence, option 'C' is correct.Maths

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