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Math Help - distance

  1. #1
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    distance

    find the distance from the origin to the point R(m^2-n^2 , 2mn) in simplest form

    i got to m^2+2mn-n^2
    but i think thats wrong

    the answer is m^2 + n^2
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  2. #2
    MHF Contributor red_dog's Avatar
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    OR=\sqrt{(m^2-n^2)^2+4m^2n^2}=\sqrt{m^4+n^4+2m^2n^2}=

    =\sqrt{(m^2+n^2)^2}=m^2+n^2
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  3. #3
    Member eXist's Avatar
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    I'm a little late but, this picture might help you as well :
    Apply the Pythagorean theorem as red dog did and you get the same answer.

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  4. #4
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    how do you go from 4m^2n^2 to then having 2m^2n^2??
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  5. #5
    Member eXist's Avatar
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    I took red dog's work and expanded it for you. Credits go to him, it's just a few more steps so you can see what's going on.

    \sqrt{(m^2-n^2)^2+4m^2n^2}

    =\sqrt{(m^2 - n^2)(m^2 - n^2) + 4m^2n^2}

    =\sqrt{m^4 -2m^2n^2 + n^4 + 4m^2n^2}

    =\sqrt{m^4+n^4+2m^2n^2}

    =\sqrt{(m^2+n^2)^2}=m^2+n^2
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  6. #6
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    Quote Originally Posted by eXist View Post
    I took red dog's work and expanded it for you. Credits go to him, it's just a few more steps so you can see what's going on.

    \sqrt{(m^2-n^2)^2+4m^2n^2}

    =\sqrt{(m^2 - n^2)(m^2 - n^2) + 4m^2n^2}

    =\sqrt{m^4 -2m^2n^2 + n^4 + 4m^2n^2}

    =\sqrt{m^4+n^4+2m^2n^2}

    =\sqrt{(m^2+n^2)^2}=m^2+n^2
    thanks heaps
    i get most of it now
    the only thing is how does the 2m^2n^2 go away??
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  7. #7
    Member eXist's Avatar
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    Well you can combine -2m^2n^2 with 4m^2n^2 since both terms have the same variables to the same power. There for you add the coefficients:

     (-2 + 4)m^2n^2 = 2m^2n^2

    From there we have:

    \sqrt{m^4 + n^4 + 2m^2n^2}

    All we are doing from here is factoring that into:

    \sqrt{(m^2 + n^2)^2}

    You can also use substitution if you can't see it right away. Ex: Let m^2 = a  n^2 = b

    That gets you:

    \sqrt{a^2 + 2ab + b^2} = \sqrt{(a + b)^2} then resubstitute m and n back in: \sqrt{(m^2 + n^2)^2}

    Either way, you come up with the same answer.
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  8. #8
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    ok i get it now
    thanks heaps
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