If-the-points-1-2i-and-1-4i-are-reflections-of-each-other-in-the-line-z-1-i-z-1-i-K-0-then-the-value-of-K-is-

Question Number 140444 by EnterUsername last updated on 07/May/21

$$\mathrm{If}\:\mathrm{the}\:\mathrm{points}\:\mathrm{1}+\mathrm{2}{i}\:\mathrm{and}\:−\mathrm{1}+\mathrm{4}{i}\:\mathrm{are}\:\mathrm{reflections}\:\mathrm{of} \\ $$$$\mathrm{each}\:\mathrm{other}\:\mathrm{in}\:\mathrm{the}\:\mathrm{line}\:{z}\left(\mathrm{1}+{i}\right)+\bar {{z}}\left(\mathrm{1}−{i}\right)+{K}=\mathrm{0},\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{K}\:\mathrm{is}\:\_\_\_\_\_. \\ $$

Answered by mr W last updated on 08/May/21

$${midpoint}\:{from}\:\mathrm{1}+\mathrm{2}{i}\:{and}\:−\mathrm{1}+\mathrm{4}{i}\:{is}\:\mathrm{3}{i}, \\ $$$${which}\:{lies}\:{on}\:{the}\:{line}. \\ $$$${z}=\mathrm{3}{i},\:\bar {{z}}=−\mathrm{3}{i} \\ $$$$\mathrm{3}{i}\left(\mathrm{1}+{i}\right)−\mathrm{3}{i}\left(\mathrm{1}−{i}\right)+{k}=\mathrm{0} \\ $$$$\mathrm{3}{i}−\mathrm{3}−\mathrm{3}{i}−\mathrm{3}+{k}=\mathrm{0} \\ $$$$\Rightarrow{k}=\mathrm{6} \\ $$

Commented by EnterUsername last updated on 12/May/21

$$\mathrm{Thanks} \\ $$

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