What is the rectangular equivalence to the parametric equations?

x (θ) = 3cosθ+2

y (θ) = 2sinθ-1,

where 0≤θ<2π.

For x (θ) = 3cosθ + 2

y² = [9x² / (x - 2) ²] - x²

For x (θ) = 3cosθ + 2

x² = [4y² / (y - 1) ²] - y²

Step-by-step explanation:

Given the following equivalence:

x² + y² = r²

r = √ (x² + y²)

x = rcosθ

cosθ = x/r

y = rsinθ

sinθ = y/r

Applying these to the given equations,

x (θ) = 3cosθ + 2

x = 3 (x/r) + 2

xr = 3x + 2r

(x - 2) r = 3x

r = 3x / (x - 2)

Square both sides

r² = 9x² / (x - 2) ²

(x - 2) ²r² = 9x²

(x - 2) ² (x² + y²) = 9x²

(x² + y²) = 9x² / (x - 2) ²

y² = [9x² / (x - 2) ²] - x²

y (θ) = 2sinθ - 1

y = 2y/r - 1

yr = 2y - r

(y - 1) r = 2y

r = 2y / (y - 1)

Square both sides

r² = 4y² / (y - 1) ²

x² + y² = 4y² / (y - 1) ²

x² = [4y² / (y - 1) ²] - y²