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Homework Help: 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

    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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    Science Advisor
    Homework Helper

    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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