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Question Number 19609 by ajfour last updated on 13/Aug/17

Answered by ajfour last updated on 13/Aug/17

Equation of L_1 : z_1 ^� z+z_1 z^� −2∣z_1 ∣^2 =0 Equation of L_2 : z_2 ^� z+z_2 z^� −2∣z_2 ∣^2 =0 and z_0 =z_1 +z_2 As z_0 lies on both lines, z_1 ^� (z_1 +z_2 )+z_1 (z_1 ^� +z_2 ^� )−2∣z_1 ∣^2 =0 ⇒∣z_1 ∣^2 +z_1 ^� z_2 +∣z_1 ∣^2 +z_1 z_2 ^� −2∣z_1 ∣^2 =0 or z_1 z_2 ^� +z_1 ^� z_2 =0 .

$$\mathrm{Equation}\:\mathrm{of}\:\mathrm{L}_{\mathrm{1}} : \\ $$$$\:\:\:\:\:\:\bar {\mathrm{z}}_{\mathrm{1}} \mathrm{z}+\mathrm{z}_{\mathrm{1}} \bar {\mathrm{z}}−\mathrm{2}\mid\mathrm{z}_{\mathrm{1}} \mid^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{Equation}\:\mathrm{of}\:\mathrm{L}_{\mathrm{2}} : \\ $$$$\:\:\:\:\:\:\bar {\mathrm{z}}_{\mathrm{2}} \mathrm{z}+\mathrm{z}_{\mathrm{2}} \bar {\mathrm{z}}−\mathrm{2}\mid\mathrm{z}_{\mathrm{2}} \mid^{\mathrm{2}} =\mathrm{0} \\ $$$$\:\mathrm{and}\:\:\:\:\:\:\mathrm{z}_{\mathrm{0}} =\mathrm{z}_{\mathrm{1}} +\mathrm{z}_{\mathrm{2}} \\ $$$$\mathrm{As}\:\mathrm{z}_{\mathrm{0}} \:\mathrm{lies}\:\mathrm{on}\:\mathrm{both}\:\mathrm{lines}, \\ $$$$\:\bar {\mathrm{z}}_{\mathrm{1}} \left(\mathrm{z}_{\mathrm{1}} +\mathrm{z}_{\mathrm{2}} \right)+\mathrm{z}_{\mathrm{1}} \left(\bar {\mathrm{z}}_{\mathrm{1}} +\bar {\mathrm{z}}_{\mathrm{2}} \right)−\mathrm{2}\mid\mathrm{z}_{\mathrm{1}} \mid^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow\mid\mathrm{z}_{\mathrm{1}} \mid^{\mathrm{2}} +\bar {\mathrm{z}}_{\mathrm{1}} \mathrm{z}_{\mathrm{2}} +\mid\mathrm{z}_{\mathrm{1}} \mid^{\mathrm{2}} +\mathrm{z}_{\mathrm{1}} \bar {\mathrm{z}}_{\mathrm{2}} −\mathrm{2}\mid\mathrm{z}_{\mathrm{1}} \mid^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{or}\:\:\:\:\mathrm{z}_{\mathrm{1}} \bar {\mathrm{z}}_{\mathrm{2}} +\bar {\mathrm{z}}_{\mathrm{1}} \mathrm{z}_{\mathrm{2}} =\mathrm{0}\:\:. \\ $$

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