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When will the line intercept the ellipse a second time?

  1. Apr 25, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the equation of the line perpendicular to the ellipse x^2 − xy + y^2 = 3 at the point (-1,1). Where does the perpendicular line intercept the ellipse a second time?


    2. Relevant equations
    ???


    3. The attempt at a solution
    I have already found the equation of the perpendicular line (f(x) = x - 2), just not sure how prove mathematically where it intercepts again. Of course I can just look at the graph, but I should be able to do it without referring to the graph at all. The slope of the line BTW is -1.
     
  2. jcsd
  3. Apr 26, 2009 #2

    D H

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    Have you tried drawing a picture?

    Does f(x)=x-2 have a slope of -1? (In other words, your perpendicular is not correct.)

    Instead of f(x), use y. Now you have two equations in two unknowns.
     
  4. Apr 26, 2009 #3
    Sorry, I just realized I made a mistake. The equation for the line PARALLEL to (-1,1) is f(x)=x-2. Sorry.

    The equation for the perpendicular line is simply f(x) = -x

    That said, since the perpendicular line passes through (-1,1) and then through the origin, you know that it intercepts the ellipse again at (1,-1). So I know the answer, just not sure how to justify it that mathematically.

    My best justification would be, "Since every point on an ellipse has a corresponding point in which the x and y values are reversed, then we know that the perpendicular line passes again through the ellipse at (1, -1)."

    Something like that, but again it would be better if I could prove this via calculations and not sentences.
     
  5. Apr 26, 2009 #4

    HallsofIvy

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    You know the perpendicular line is y= -x and you want to determine where it crosses x2[/sub]- xy+ y2= 3. Okay, replace each y with -x to get x2- x(-x)+ (-x)2= 3x2= 3 and solve for x. You already know one solution is x= -1 since (-1, 1) is on the ellipse. What is the other solution?
     
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