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Math Help - Tangent lines to an ellipse from an external point

  1. #1
    Junior Member Kaloda's Avatar
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    Tangent lines to an ellipse from an external point

    Given an the equation of an ellipse, say \frac{x^2}{9}+\frac{y^2}{16}=1,
    find the equations of the two lines tangent to this ellipse passing through the external point
    P(5,6). Thank you in advance.
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Tangent lines to an ellipse from an external point

    Quote Originally Posted by Kaloda View Post
    Given an the equation of an ellipse, say \frac{x^2}{9}+\frac{y^2}{16}=1,
    find the equations of the two lines tangent to this ellipse passing through the external point
    P(5,6). Thank you in advance.
    1. Solve the equation for y^2
    2. Implicit differentiation

    That gives the equation for the tangent lines.
    3. Plug in point P.
    4. What can you come up with?

    -Dan
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  3. #3
    Member kalyanram's Avatar
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    Re: Tangent lines to an ellipse from an external point

    Compute the slope of the tangent at the point of contact say (\alpha,\beta)which is \frac{dy}{dx} = \frac{-16\alpha}{9\beta}. Now we have the slope of the line also as \frac{6-\beta}{5-\alpha} = \frac{-16\alpha}{9\beta} \implies 16\alpha^2 + 9 \beta^2 = 9*6\beta+9*10\alpha from the equation of ellipse we have 16\alpha^2 + 9 \beta^2 = 9*16=9*6\beta+9*10\alpha \implies 5\alpha+3\beta=8 substitute in \frac{\alpha^2}{9} + \frac{\beta^2}{16} = 1 and solve for \beta you will get two points on the ellipse, and hence you can get two lines.

    ~Kalyan.
    Attached Thumbnails Attached Thumbnails Tangent lines to an ellipse from an external point-ellipse.png  
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  4. #4
    Junior Member Kaloda's Avatar
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    Re: Tangent lines to an ellipse from an external point

    I had already figured it out but THANKS.
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