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12 March, 14:55

Find a cartesian equation for the curve. r = 9 sin (θ)

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  1. 12 March, 15:18
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    For this case we have the following equation:

    r = 9 sin (θ)

    In addition, we have the following change of variables:

    y = r * sine (θ)

    Rewriting the equation we have:

    r = 9 sin (θ)

    r = 9 (y / r)

    r ^ 2 = 9y

    On the other hand:

    r ^ 2 = x ^ 2 + y ^ 2

    Substituting values:

    x ^ 2 + y ^ 2 = 9y

    Rewriting:

    x ^ 2 + y ^ 2 - 9y = 0

    Completing squares:

    x ^ 2 + y ^ 2 - 9y + (-9/2) ^ 2 = (-9/2) ^ 2

    Rewriting:

    x ^ 2 + 1/4 (2y-9) ^ 2 = 81/4

    4x ^ 2 + (2y-9) ^ 2 = 81

    Answer:

    The Cartesian equation is:

    4x ^ 2 + (2y-9) ^ 2 = 81
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