Question

r = 2/(r-sin(theta)) convert the given equation to a rectangular equation

Answer 1

The connection between polar and cartesian coordinates is:\[x=r \cos \theta\]\[y=r \sin \theta\]or\[r^2=x^2+y^2\]\[\theta=\arctan \frac{ y }{ x }\]
Hint: first rewrite the equation as:\[r(r-\sin \theta)=2\]then \[r^2-r \sin \theta=2\]then do some substitutions...
Just ask if you don't succeed!

Answer 2

If you substitute the values for r^2 and rsin(theta) as stated above, you get:
\[x^2+y^2-y=2\]This is already fine, because it is a rectangular equation. But there is one more step. Complete the square for y:\[x^2+(y-\frac{ 1 }{ 2 })^2-\frac{ 1 }{ 4 }=2\]Or:\[x^2+(y-\frac{ 1 }{ 2 })^2=2\frac{ 1 }{ 4 }\]This is the equation of a circle with center (0, 1/2) and radius 3/2! So it is a very simple curve after all...