P-is-the-point-representing-the-complex-number-z-r-cos-i-sin-in-an-argand-diagram-such-that-z-a-z-a-a-2-Show-that-P-moves-on-the-curve-whose-equation-is-r-2-2a-2-cos2-sketch-t

Tinku Tara June 4, 2023 Others 0 Comments

Question Number 94081 by Rio Michael last updated on 16/May/20

P is the point representing the complex number z = r( cos θ + i sin θ) in an argand diagram such that ∣z−a∣∣z + a∣ = a^2 . Show that P moves on the curve whose equation is r^2 =2a^2 cos2θ. sketch the curve r^2 = 2a^2 cos 2θ , showing clearly the tangents at the pole.

$${P}\:\mathrm{is}\:\mathrm{the}\:\mathrm{point}\:\mathrm{representing}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{number} \\ $$$$\:{z}\:=\:{r}\left(\:\mathrm{cos}\:\theta\:+\:{i}\:\mathrm{sin}\:\theta\right)\:\mathrm{in}\:\mathrm{an}\:\mathrm{argand}\:\mathrm{diagram}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mid{z}−{a}\mid\mid{z}\:+\:{a}\mid\:=\:{a}^{\mathrm{2}} .\:\mathrm{Show}\:\mathrm{that}\:{P}\:\mathrm{moves}\:\mathrm{on}\:\mathrm{the}\:\mathrm{curve} \\ $$$$\mathrm{whose}\:\mathrm{equation}\:\mathrm{is}\:{r}^{\mathrm{2}} \:=\mathrm{2}{a}^{\mathrm{2}} \:\mathrm{cos2}\theta.\:\mathrm{sketch}\:\mathrm{the}\:\mathrm{curve}\: \\ $$$${r}^{\mathrm{2}} \:=\:\mathrm{2}{a}^{\mathrm{2}} \:\mathrm{cos}\:\mathrm{2}\theta\:,\:\mathrm{showing}\:\mathrm{clearly}\:\mathrm{the}\:\mathrm{tangents}\:\mathrm{at}\:\mathrm{the}\:\mathrm{pole}. \\ $$

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