Grade 12th passDifferential Calculus

find the orthogonal trajectory of cardiods r^2=a^2sin2theta

20KT1A04F3 Jagadeesh

5 Years agoGrade 12th pass

1 Answer

khushi garg

5 Years ago

r²=a²sin² $\theta$ $\rightarrow$ 1

now derivative wrt $\theta$ where a is constant

2rdr/d $\theta$ =a²(2sin $\theta$ cos $\theta$ ) $\rightarrow$ 2

now put value of a² from 1 equation to 2 equation

a²=r²/sin² $\theta$

2rdr/d $\theta$ =(r²/sin² $\theta$ )(2sin $\theta$ cos $\theta$ )

dr/d $\theta$ =(2r²sin $\theta$ cos $\theta$ )/(2rsin² $\theta$ )

dr/d $\theta$ =(rcos $\theta$ )/(sin $\theta$ )

dr/d $\theta$ =rcot $\theta$

now replace dr/d $\theta$ with -r²d $\theta$ /dr

-r²d $\theta$ /dr=rcot $\theta$

d $\theta$ /dr=-cot $\theta$ /r

-d $\theta$ /cot $\theta$ =dr/r

now integrate both side

-∫tan $\theta$ d $\theta$ =∫dr/r

ln(cos $\theta$ )=ln(r)+ln(C)

cos $\theta$ =rc

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find the orthogonal trajectory of cardiods r^2=a^2sin2theta

20KT1A04F3 Jagadeesh

1 Answer

khushi garg

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