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Thread: tangent line, implicit differentiation

  1. #1
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    tangent line, implicit differentiation

    Find the equations of the tangents to the ellipse $\displaystyle x^{2} + 4y^{2} = 4 $

    which are perpendicular t0 the line $\displaystyle 2x-3y = 1 $


    I am not sure how to work out the equations of the tangents, when I have not been given a specified point,

    $\displaystyle \frac{dy}{dx} = -\frac{x}{4y} $

    $\displaystyle y = \frac{2}{3}x - \frac{1}{3} $

    And if the tangents are perpendicular to the line, than it has to have a gradient of $\displaystyle -\frac{3}{2} $

    that's all I am able to do, any help appreciated.

    thank you.
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  2. #2
    Super Member Random Variable's Avatar
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    then $\displaystyle -\frac{x}{4y} = - \frac{3}{2} $

    or $\displaystyle x = 6y$


    so $\displaystyle (6y)^{2} + 4 y^{2} = 4 $

    or $\displaystyle y = \pm \frac{1}{\sqrt{10}} $

    now just plug back in the y-values to find the x-values
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