by "Cartesian", I assume rectangular or (x,y).

x = r*cosT ------------cosT = x/r

y = r*sinT-----------------sinT = y/r

r = sqrt(x^2 +y^2)

1) r*cosT = 1

That is x = 1 -----------a vertical line, answer.

2) r = tanT*secT

r = sinT/cosT *1/cosT

r = sinT / cos^2(T)

r = (y/r) / (x/r)^2

r = ry / x^2

1 = y / x^2

x^2 = y

y = x^2 -------a vertical parabola that opens upward, answer.

3) xy = 4

(r*cosT)(r*sinT) = 4

(r^2)(sinTcosT) = 4

(r^2)((1/2)sin(2T)) = 4

(r^2)sin(2T) = 8

r^2 = 8/sin(2T)

r^2 = 8csc(2T)

r = 2sqrt[2csc(2T)] -----------answer.