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Math Help - implicit differentiation question

  1. #1
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    implicit differentiation question

    find the equations of the lines that are tangent to the eclipse x^2 + 4y^2 =16 and that also pass through the point (4,6)

    i implicitly differentiated it and i got -x/4y

    then i plugged the point (4,6) to find the slope and got an equation that is 6y + x -40 = 0 but according to the text the answers are x-4=0 and 2x-3y + 10 = 0

    what did *I do wrong and how do i solve this problem using implicit differentiation.

    edit: typo
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  2. #2
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    Re: implicit differentiation question

    The derivative gives the slope of the the curve at any point *on the curve*. The point (4,6) is not on the curve.

    For any point (x_0,y_0) on the ellipse, you can calculate the tangent line as y-y_0 = -\frac{x_0}{4y_0}(x-x_0). Plugging in (x,y)=(4,6) gives you y_0 as a function of x_0. You can then plug this into x_0^2 + 4y_0^2 =16 to get a quadratic equation in x_0. The two solutions should give you the two tangent lines.

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