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Points on ellipse where tangent slope=1

  1. Apr 3, 2013 #1
    1. The problem statement, all variables and given/known data

    7
    Find the points on the ellipse x^2 + 2y^2 = 1 where the tangent line has slope 1

    2. Relevant equations



    3. The attempt at a solution
    I got the correct X and Y values but this gives me four possibilities and the answer key says there are two points.

    I got x = +/- sqrt 2/3 and y = +/- sqrt 1/6
    The values are correct but how am I supposed to know which coordinate to pare with which?

    For an example, how do I know if it's (-sqrt 2/3, -sqrt 1/6) or (sqrt 2/3, -sqrt 1/6) ?
     
  2. jcsd
  3. Apr 3, 2013 #2
    It helps to draw a diagram. If the slope is +1, aren't there only two possible tangents? In which quadrants do their points of intersection with the ellipse lie?
     
  4. Apr 3, 2013 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    From [itex]x^2+ 2y^2= 1[/itex] you get 2x+ 2yy'= 0 so that y'=- x/y= 1. That gives y= -x and putting that into the equation of the ellipse, [itex]x^2+ 2x^2= 3x^2= 1[/itex] so that [itex]x= \pm\sqrt{3}/3[/itex].

    That's what you got , right? And that gives four points on the ellipse?

    But your condition is y= -x. Putting [itex]x= \sqrt{3}/3[/itex] and [itex]x= -\sqrt{3}/3[/itex] into that gives only two points.

    (The other two points on the ellipse give slope -1.)
     
  5. Apr 3, 2013 #4
    Oh, DUH! Thanks guys :D.
     
  6. Apr 3, 2013 #5

    Mark44

    Staff: Mentor

    It's hard to overemphasize the importance of this advice. A halfway decent diagram can give you insight that you can't get just fiddling with equations.
     
  7. Apr 3, 2013 #6
    I actually did draw a diagram the first time I did it but I still got four points, it just didn't strike me to draw the tangent lines. If this question is on my test today (in 1.5hrs) I'll nail it. Thanks again.
     
  8. Apr 3, 2013 #7

    Mark44

    Staff: Mentor

    Try to draw a graph that represents the situation; namely, an ellipse with some points where the slope of the tangent line equals 1. If you got four points, your drawing wasn't an accurate representation of the question.
     
  9. Apr 3, 2013 #8
    I realize that now. It became painfully obvious as soon as I read Mr Simha's answer.
     
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