1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    User Avatar
    Staff Emeritus
    Science Advisor

    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


    User Avatar
    Science Advisor

    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook