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Math Help - Tangent to the ellipse

  1. #1
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    Tangent to the ellipse

    I need help with this maths question...

    Prove that if the line lx +my + n = 0 touches the ellipse b^2 x^2 + a^2 y^2 = a^2 b^2 then a^2 l^2 + b^2 m^2 = n^2

    I tried substituting the first equation in terms of x or y, then substitute into the second equation but I ended up having too many unknowns.... any ideas how to solve this question...

    Thank you..
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  2. #2
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    Quote Originally Posted by ecogreen View Post
    I need help with this maths question...

    Prove that if the line lx +my + n = 0 touches the ellipse b^2 x^2 + a^2 y^2 = a^2 b^2 then a^2 l^2 + b^2 m^2 = n^2

    I tried substituting the first equation in terms of x or y, then substitute into the second equation but I ended up having too many unknowns.... any ideas how to solve this question...

    Thank you..
    1. Calculate the coordinates of the points of intersection. If the line is actually a tangent to the ellipse you only get one point of intersection.

    2. lx+my+n=0~\implies~y=-\dfrac{lx+n}{m}

    Plug in the term for y into the 2nd equation and solve for x.

    3. b^2 x^2+a^2 \cdot \left(-\dfrac{lx+n}{m}\right)^2=a^2 b^2

    Expand the bracket and collect like terms:

    \dfrac{a^2 l^2 +b^2 m^2}{m^2} \cdot x^2+\dfrac{2 a^2 l n}{m^2} \cdot x + \dfrac{a^2 n^2}{m^2} = a^2 b^2

    This is a quadratic equation in x. Use the quadratic formula and solve for x:
    x = \dfrac{-a^2 l n \pm \sqrt{a^2 l^2 + b^2 m^2 - n^2}  }{a^2 l^2 + b^2 m^2}

    4. You only get one common point of the line and the ellipse (= tangent point) if the radical equals zero:

    a^2 l^2 + b^2 m^2 - n^2=0~\implies~\boxed{a^2 l^2 + b^2 m^2 = n^2}
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