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Math Help - Where does the perpendicular line intersect the ellipse again?

  1. #1
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    Where does the perpendicular line intersect the ellipse again?

    I have a question which asks me to find the equation of a line perpendicular to the ellipse x^2-xy+y^2=3, at the point (-1, 1). I did this by differentiating implicitly, finding the tangent line then the line perpendicular to that. Pretty simple. The equation of the perpendicular line turned out to be f(x)=-x.

    The question now is where does this line intersect the ellipse again? I know the answer is (1, -1) by looking at the graph, but I need to justify this mathematically. Anyone know any algebra gymnastics which will lead me to this answer, based only on f(x)=-x, and the point (-1, 1)?
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  2. #2
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    Hello, Ares_D1!

    A slight change in notation should clear it up . . .


    Find the equation of a line perpendicular to the ellipse x^2-xy+y^2\:=\:3, at the point (-1, 1).

    The equation of the perpendicular line turned out to be: . {\color{blue}y \:=\:x}

    The question now is where does this line intersect the ellipse again?

    We want the intersection of two curves: the ellipse and the line.

    Substitute y = x into: . x^2-xy+y^2\:=\:3 . . . and solve.

    We'll get two points . . . and we already know one of them . . . Got it?

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  3. #3
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    The equation of the perpendicular is y= -x, not y= x.
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