Find foci,directrices,, vertisec and find cartesian of the conic r=3/(2-cost)

r = 3/(2−cos t)
2r − r cos t = 3
2√(x²+y²) − x = 3
2√(x²+y²) = x + 3

Square both sides:
4x² + 4y² = x² + 6x + 9
3x² − 6x + 4y² = 9
3(x² − 2x + 1) + 4y² = 9 + 3(1)
3 (x − 1)² + 4y² = 12
(x − 1)²/4 + y²/3 = 1

a = 2
b = √3
c² = a² − b² = 4 − 3 = 1
c = 1

Centre: (1, 0)
Vertices: (1±a, 0) = (−1, 0) and (3, 0)
Foci: (1±c, 0) = (0, 0) and (2, 0)

Ellipse does not have a directrix