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Answer 1 · 2018-02-01T02:56:30

#color(blue)(r=-8sin(theta))#

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From the diagram we can see that point P has polar coordinates
#( r , theta )# and Cartesian coordinates #(x,y)#.

And #color(white)(88)x=rcos(theta)# , #y = rsin(theta)#

#(x,y) -> (rcos(theta), rsin(theta))#

Also:

By Pythagoras' Theorem :

#r^2=(rcostheta)^2+(rsintheta)^2#

Since:

#x=rcos(theta)# and #y = rsin(theta)#

Then:

#r^2=x^2+y^2# #:. r=sqrt(x^2+y^2)#

Using these ideas:

#x^2+y^2+8y=0#

From above:

#x^2+y^2=r^2#

#y=rsin(theta)#

Substituting in #color(white)(888)x^2+y^2+8y=0#

#r^2+8rsin(theta)=0#

Dividing by #r#:

#r+8sin(theta)=0#

#color(blue)(r=-8sin(theta))#