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5 April, 08:51

What is the rectangular equivalence to the parametric equations?

x (θ) = 3cosθ+2

y (θ) = 2sinθ-1,

where 0≤θ<2π.

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  1. 5 April, 09:04
    0
    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
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