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Question

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


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

161523_157173_ans.JPG

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