How do you convert #r = 12/(4 + 8 sinx)# into a rectangular equation?

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Answer 1 · 2015-11-22T14:30:10

Use these 3 identities to solve polar equations:

1) #r=sqrt(x^2+y^2)#
2) #x=rcostheta#
3) #y=rsintheta#

First simplify the polar equation:

#r=3/(1+2sintheta)#

#r+2rsintheta=3#

Using the identities above, use substitution to eliminate the r and sine function:

#sqrt(x^2+y^2)+(2r)(y/r)=3#

Now, simplify

#sqrt(x^2+y^2)+2y=3#

#sqrt(x^2+y^2)=3-2y#

#x^2+y^2=(3-2y)^2=9-12y+4y^2#

Rectangular Equation : #3y^2-x^2-12y+9=0#

This is a hyperbola with a vertical transverse axis .

Hope that helped

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