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Question Number 136988 by mohammad17 last updated on 28/Mar/21

whats the intersiction of the curves r=(1/(1−cosθ)) and r=(1/(1+cosθ)) ?

$${whats}\:{the}\:{intersiction}\:{of}\:{the}\:{curves}\:{r}=\frac{\mathrm{1}}{\mathrm{1}−{cos}\theta} \\ $$$$ \\ $$$${and}\:{r}=\frac{\mathrm{1}}{\mathrm{1}+{cos}\theta}\:? \\ $$

Answered by Ñï= last updated on 28/Mar/21

r=(1/(1−cos θ)) ⇒r−rcos θ=(√(x^2 +y^2 ))−x=1 (1) r=(1/(1+cos θ)) ⇒r+rcos θ=(√(x^2 +y^2 ))+x=1 (2) (2)∧(1) ⇒x=0 y=±1 intersect points:(0,1),(0,−1)

$${r}=\frac{\mathrm{1}}{\mathrm{1}−\mathrm{cos}\:\theta} \\ $$$$\Rightarrow{r}−{r}\mathrm{cos}\:\theta=\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }−{x}=\mathrm{1}\:\:\:\left(\mathrm{1}\right) \\ $$$${r}=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cos}\:\theta} \\ $$$$\Rightarrow{r}+{r}\mathrm{cos}\:\theta=\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }+{x}=\mathrm{1}\:\:\:\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{2}\right)\wedge\left(\mathrm{1}\right) \\ $$$$\Rightarrow{x}=\mathrm{0}\:\:{y}=\pm\mathrm{1} \\ $$$${intersect}\:{points}:\left(\mathrm{0},\mathrm{1}\right),\left(\mathrm{0},−\mathrm{1}\right) \\ $$

Answered by physicstutes last updated on 29/Mar/21

At the point of intersection,these two polar curves are equal. That is (1/(1− cos θ)) = (1/(1 + cos θ)) ⇒ 1−cos θ = 1 + cos θ cos θ = 0 or θ = 2πn±(π/2) hence θ={(π/2),−(π/2)} points of intersection are (1,(π/2)) and (−1,−(π/2))

$$\mathrm{At}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{intersection},\mathrm{these}\:\mathrm{two} \\ $$$$\mathrm{polar}\:\mathrm{curves}\:\mathrm{are}\:\mathrm{equal}.\:\mathrm{That}\:\mathrm{is} \\ $$$$\:\frac{\mathrm{1}}{\mathrm{1}−\:\mathrm{cos}\:\theta}\:=\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{cos}\:\theta} \\ $$$$\Rightarrow\:\mathrm{1}−\mathrm{cos}\:\theta\:=\:\mathrm{1}\:+\:\mathrm{cos}\:\theta \\ $$$$\mathrm{cos}\:\theta\:=\:\mathrm{0}\:\mathrm{or}\:\theta\:=\:\mathrm{2}\pi{n}\pm\frac{\pi}{\mathrm{2}} \\ $$$$\mathrm{hence}\:\theta=\left\{\frac{\pi}{\mathrm{2}},−\frac{\pi}{\mathrm{2}}\right\} \\ $$$$\mathrm{points}\:\mathrm{of}\:\mathrm{intersection}\:\mathrm{are}\:\left(\mathrm{1},\frac{\pi}{\mathrm{2}}\right)\:\mathrm{and}\:\left(−\mathrm{1},−\frac{\pi}{\mathrm{2}}\right) \\ $$$$ \\ $$

Commented by physicstutes last updated on 29/Mar/21

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