Find the Cartesian equation for the curve?

Question

Find the Cartesian equation for the curve described by the polar equation r=1/(1-sin(theta)).

Answer 1

Use polar coordinates:     x = rcosθ    y = rsinθ       which gives us

                                                   r = sqrt (x² + y²) 

In terms of x and y    sinθ  = y/ sqrt(x² +y²). Thus, the initial equation can be rewritten in the following way:

                                  1 = sqrt(x² +y²) - y       or     1+ y = sqrt(x² +y²)

Raising both sides into square we will obtain   

                                                y = (1/2)(x² -1)

Answer 2

r=1/(1-sinθ)

=> r(1-sinθ) = 1

=> r - rsinθ = 1

=> r = 1+rsinθ

Square both sides,

r^2 = 1+2rsinθ + (rsinθ)^2

x^2+y^2 = 1+2y+y^2, since r^2 = x^2+y^2, rsinθ = y.

Solve for y,

y = (1/2)x^2 - (1/2) <==Answer