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Simplify this expression for an ellipse

  1. Mar 16, 2014 #1
    1. The problem statement, all variables and given/known data

    Simplify sqrt(x^2+(y-sqrt(5))^2) = 8 - sqrt(x^2 + (y+sqrt(5))^2)

    3. The attempt at a solution

    I know squaring both sides, collecting like terms and simplifying gets the equation but in my solution manual they do it a different way that is a lot shorter and I need help understanding what they did:

    sqrt(x^2+(y-sqrt(5))^2) = 8 - sqrt(x^2 + (y+sqrt(5))^2)
    16 + y*sqrt(5) = 4*sqrt(x^2 +(y+sqrt(5))^2)
    from this step ^^ to the following is where I am having trouble
    16x^2 + 11y^2 = 176

    Any help will be greatly appreciated. Thank you!
     
  2. jcsd
  3. Mar 16, 2014 #2
    They square the left hand side and right hand side again.
     
  4. Mar 16, 2014 #3
    But if you square the left and right side wont the trinomial on the right side expand to something absurbdly long?
     
  5. Mar 16, 2014 #4
    On the right is four times the square root of something, so when you square it you get 16(x^2 +(y+sqrt(5))^2)

    What's the problem?
     
  6. Mar 16, 2014 #5
    Oh yes sorry I believe I pointed to the wrong line. I am having trouble going from this line:

    sqrt(x^2+(y-sqrt(5))^2) = 8 - sqrt(x^2 + (y+sqrt(5))^2)

    to this line:

    16 + y*sqrt(5) = 4*sqrt(x^2 +(y+sqrt(5))^2)

    Sorry for the mistake and thank you again!
     
  7. Mar 16, 2014 #6
    After squaring both sides once you get
    $$x^2+(y-\sqrt{5})^2=64-16\sqrt{x^2+(y+\sqrt{5})^2}+x^2+(y+\sqrt{5})^2$$

    Then put the square root on one side on its own, and all the other terms on the other side, and square again.
     
  8. Mar 16, 2014 #7

    Ray Vickson

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    For ##f_1 = \sqrt{x^2+(y-\sqrt{5})^2}## and ##f_2 = \sqrt{x^2+(y+\sqrt{5})^2}## we can write the equation as ##f_1+f_2 = 8##, hence ##64 = (f_1 + f_2)^2 = f_1^2 + f_2^2 + 2 f_1 f_2##. Re-write this as ##2 f_1 f_2 = 64 - f_1^2 - f_2^2 \equiv R##, and note that when you expand out and simplify the right-hand-side R it does not contain any square roots at all. Square again to get ##4 f_1^2 f_2^2 = R^2##.
     
  9. Mar 16, 2014 #8
    Yes, that is exactly what I did, but I wanted to know how to get this line that was given in the solution manual from the original expression:

    16 + y*sqrt(5) = 4*sqrt(x^2 +(y+sqrt(5))^2)
     
  10. Mar 16, 2014 #9
    Can you show us your next steps so we can see where you went wrong?
     
  11. Mar 16, 2014 #10
    Actually I understand what you meant. I redid the problem square it twice like you said and got the desired equation. Thank you all for your help and sorry for the confusion!
     
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