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Math Help - Help with cartesian equation

  1. #1
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    Help with cartesian equation

    x = 2 cot t
    y = 2 sin^2 t

    I need to find the cartesian equation in the form y=f(x). Can somebody pls help
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  2. #2
    MHF Contributor
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    Hi

    Use xē = cotēt = cosēt / sinēt = (1-sinēt)/sinēt
    substitute sinēt=y/2
    and finally rearrange to get y=f(x)
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  3. #3
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    thanks you're such a lifesaver
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  4. #4
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    Hello, threerose!

    Here's another approach.
    . . It should surprise/impress/terrify your professor.


    \begin{array}{c}x \:=\: 2\cot\theta \\<br />
y \:=\: 2\sin^2\theta \end{array}

    Find the cartesian equation in the form y\,=\,f(x)

    We have: . \begin{array}{cccccccc}<br />
x \;=\;2\cot\theta &\Longrightarrow& \cot\theta \;=\; \dfrac{x}{2} &\Longrightarrow& \cot^2\theta \;=\; \dfrac{x^2}{4} & [1] \\ \\[-3mm]<br />
y \;=\; 2\sin^2\theta &\Longrightarrow& y \;=\; \dfrac{2}{\csc^2\theta} &\Longrightarrow & \csc^2\theta \;=\; \dfrac{2}{y} & [2] \end{array}

    \text{Subtract }[2]\text{ - }[1]\!:\;\;\underbrace{\csc^2\!\theta - \cot^2\!\theta}_{\text{This is 1}} \;=\;\frac{2}{y} - \frac{x^2}{4}

    . . So we have: . 1 \:=\:\frac{2}{y} - \frac{x^2}{4}

    Multiply by 4y\!:\;\;4y \;=\;8 - x^2y \quad\Rightarrow\quad x^2y+4y \:=\:8 \quad\Rightarrow\quad y(x^2+4)\:=\:8


    Therefore: . y \;=\;\frac{8}{x^2+4}

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