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Math Help - Cartesian equation of tangent of ellipse

  1. #1
    Member ssadi's Avatar
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    Cartesian equation of tangent of ellipse

    Show that for all values of m, the straight lines with equations y=mx\pm \sqrt(b^2+a^2m^2) are tangents to the ellipse with equation \frac{x^2}{a^2}+\frac{y^2}{b^2}=1
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  2. #2
    MHF Contributor red_dog's Avatar
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    The system formed by the equation of the line and the equation of the ellipse has a single solution, therefore the line intersects the ellipse in a single point.
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  3. #3
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    You might find it simpler to multiply on both sides of the equation to get b^2x^2+ a^2y^2= a^2b^2 so you can use implicit differentiation: 2b^2x+ 2a^2yy'= 0 and y'= -\frac{b^2x}{a^2y}.
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