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

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

Grade:12th pass

1 Answers

khushi garg
16 Points
one year ago
r2=a2sin2\theta                   \rightarrow        1
now derivative wrt \theta  where a is constant 
2rdr/d\theta =a2(2sin\thetacos\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\thetacos\theta)
dr/d\theta=(2r2sin\thetacos\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\thetad\theta=∫dr/r
ln(cos\theta)=ln(r)+ln(C)
cos\theta=rc

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