1. Not finding help here? Sign up for a free 30min 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!

Level curves

  1. Sep 11, 2007 #1
    1. The problem statement, all variables and given/known data

    I need to sketch level curves of [itex]T(x, y) = 50(1 + x^2 + 3y^2)^{-1}[/itex] and [itex] V(x, y) = \sqrt{1 - 9x^2 -4y^2}[/itex]

    3. The attempt at a solution

    Is it correct that they are ellipses?

    ie [tex] 1 = \frac{9}{1 - c^2} x^2 + \frac{4}{1 - c^2}y^2[/itex]

    for V(x, y) = c = constant
    I feel so rusty going back to school :s
    Last edited: Sep 11, 2007
  2. jcsd
  3. Sep 11, 2007 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    That's for V? Setting [itex]V(x,y)= \sqrt{1- 9x^2- 4y^2}= c[/itex], then [itex]1- 9x^2- 4y^2= c^2[/itex], [itex]9x^2+ 4y^2= 1-c^2[/itex],
    [tex]\frac{9}{1- c^2}x^2+ \frac{4}{1-c^2}y^2= 1[/tex]
    just as you say. Yes, that's an ellipse. It might be easier to recognise if you wrote it
    [tex]\frac{x^2}{\left(\frac{\sqrt{1-c^2}}{3}\right)^2}+ \frac{y^2}{\left(\frac{\sqrt{1-c^2}}{2}\right)^2}= 1[/tex]
    an ellipse with center at (0,0) and semi-axes of length
    [tex]\frac{\sqrt{1-c^2}}{3} [/tex]

    Similarly, [itex]T(x,y)= 50(1+ x^2+ 3y^2)^{-1}= c[/itex] gives [itex]c(1+ x^2+ 3y^2)= 50[/itex] so [itex]1+ x^2+ 3y^2= 50/c[/itex], [itex]x^2+ 3y^2= (50/c- 1)[/itex]. Now divide both sides by 50/c- 1:
    [tex]\frac{x^2}{50/c-1}+ \frac{y^2}{\frac{50/c-1}{3}}= 1[/tex]
    again, an ellipse with center at (0,0), semi-axes of length
    [tex]\sqrt{50/c- 1}[/tex]
    [tex]\sqrt{\frac{50/c- 1}{3}}[/tex]
  4. Sep 11, 2007 #3
    That does make it easier to understand.

    Thanks for your help.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Level curves
  1. Level curves (Replies: 7)

  2. Level Curve (Replies: 1)

  3. Level curves? (Replies: 2)

  4. Level curves (Replies: 5)