Find the range of $xy$ under the conditions: $x^2-xy+y^2=9$ and $|x^2-y^2|<9$











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Assume $x,y in mathbb{R}^+$, and satisfied the following expression:
$$x^2-xy+y^2=9$$
$$left|x^2-y^2right| < 9$$
find the range of $xy$






My approach: $x^2-xy+y^2=9$ $Rightarrow$ $xy+9=x^2+y^2 geq 2xy$ $Rightarrow$ $xy leq 9$

But I don't know how to find the lower bound. please help me..thanks very much.










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    up vote
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    Assume $x,y in mathbb{R}^+$, and satisfied the following expression:
    $$x^2-xy+y^2=9$$
    $$left|x^2-y^2right| < 9$$
    find the range of $xy$






    My approach: $x^2-xy+y^2=9$ $Rightarrow$ $xy+9=x^2+y^2 geq 2xy$ $Rightarrow$ $xy leq 9$

    But I don't know how to find the lower bound. please help me..thanks very much.










    share|cite|improve this question
























      up vote
      1
      down vote

      favorite
      2









      up vote
      1
      down vote

      favorite
      2






      2





      Assume $x,y in mathbb{R}^+$, and satisfied the following expression:
      $$x^2-xy+y^2=9$$
      $$left|x^2-y^2right| < 9$$
      find the range of $xy$






      My approach: $x^2-xy+y^2=9$ $Rightarrow$ $xy+9=x^2+y^2 geq 2xy$ $Rightarrow$ $xy leq 9$

      But I don't know how to find the lower bound. please help me..thanks very much.










      share|cite|improve this question













      Assume $x,y in mathbb{R}^+$, and satisfied the following expression:
      $$x^2-xy+y^2=9$$
      $$left|x^2-y^2right| < 9$$
      find the range of $xy$






      My approach: $x^2-xy+y^2=9$ $Rightarrow$ $xy+9=x^2+y^2 geq 2xy$ $Rightarrow$ $xy leq 9$

      But I don't know how to find the lower bound. please help me..thanks very much.







      algebra-precalculus analysis






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      asked Nov 26 at 2:49









      Ginkgo

      82




      82






















          1 Answer
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          For the lower bound you can use the follwing facts:




          1. $x^2-xy+y^2 = (x-y)^2 +xy Rightarrow 9-xy = (x-y)^2$


          2. $x^2-xy+y^2 = (x+y)^2 -3xy Rightarrow 9+3xy = (x+y)^2$

          3. $left|x^2-y^2right| < 9 Leftrightarrow (x+y)^2(x-y)^2 < 81$


          Plugging 1. and 2. into 3. you get:
          $$(x+y)^2(x-y)^2 < 81 Leftrightarrow (9-xy)(9+3xy) < 81 Leftrightarrow xy(6-xy) < 0 stackrel{x,y >0}{Leftrightarrow} boxed{xy>6}$$
          Together with your upper bound $boxed{xy leq 9}$ you get
          $$boxed{6 < xy leq 9}$$






          share|cite|improve this answer























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            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            3
            down vote



            accepted










            For the lower bound you can use the follwing facts:




            1. $x^2-xy+y^2 = (x-y)^2 +xy Rightarrow 9-xy = (x-y)^2$


            2. $x^2-xy+y^2 = (x+y)^2 -3xy Rightarrow 9+3xy = (x+y)^2$

            3. $left|x^2-y^2right| < 9 Leftrightarrow (x+y)^2(x-y)^2 < 81$


            Plugging 1. and 2. into 3. you get:
            $$(x+y)^2(x-y)^2 < 81 Leftrightarrow (9-xy)(9+3xy) < 81 Leftrightarrow xy(6-xy) < 0 stackrel{x,y >0}{Leftrightarrow} boxed{xy>6}$$
            Together with your upper bound $boxed{xy leq 9}$ you get
            $$boxed{6 < xy leq 9}$$






            share|cite|improve this answer



























              up vote
              3
              down vote



              accepted










              For the lower bound you can use the follwing facts:




              1. $x^2-xy+y^2 = (x-y)^2 +xy Rightarrow 9-xy = (x-y)^2$


              2. $x^2-xy+y^2 = (x+y)^2 -3xy Rightarrow 9+3xy = (x+y)^2$

              3. $left|x^2-y^2right| < 9 Leftrightarrow (x+y)^2(x-y)^2 < 81$


              Plugging 1. and 2. into 3. you get:
              $$(x+y)^2(x-y)^2 < 81 Leftrightarrow (9-xy)(9+3xy) < 81 Leftrightarrow xy(6-xy) < 0 stackrel{x,y >0}{Leftrightarrow} boxed{xy>6}$$
              Together with your upper bound $boxed{xy leq 9}$ you get
              $$boxed{6 < xy leq 9}$$






              share|cite|improve this answer

























                up vote
                3
                down vote



                accepted







                up vote
                3
                down vote



                accepted






                For the lower bound you can use the follwing facts:




                1. $x^2-xy+y^2 = (x-y)^2 +xy Rightarrow 9-xy = (x-y)^2$


                2. $x^2-xy+y^2 = (x+y)^2 -3xy Rightarrow 9+3xy = (x+y)^2$

                3. $left|x^2-y^2right| < 9 Leftrightarrow (x+y)^2(x-y)^2 < 81$


                Plugging 1. and 2. into 3. you get:
                $$(x+y)^2(x-y)^2 < 81 Leftrightarrow (9-xy)(9+3xy) < 81 Leftrightarrow xy(6-xy) < 0 stackrel{x,y >0}{Leftrightarrow} boxed{xy>6}$$
                Together with your upper bound $boxed{xy leq 9}$ you get
                $$boxed{6 < xy leq 9}$$






                share|cite|improve this answer














                For the lower bound you can use the follwing facts:




                1. $x^2-xy+y^2 = (x-y)^2 +xy Rightarrow 9-xy = (x-y)^2$


                2. $x^2-xy+y^2 = (x+y)^2 -3xy Rightarrow 9+3xy = (x+y)^2$

                3. $left|x^2-y^2right| < 9 Leftrightarrow (x+y)^2(x-y)^2 < 81$


                Plugging 1. and 2. into 3. you get:
                $$(x+y)^2(x-y)^2 < 81 Leftrightarrow (9-xy)(9+3xy) < 81 Leftrightarrow xy(6-xy) < 0 stackrel{x,y >0}{Leftrightarrow} boxed{xy>6}$$
                Together with your upper bound $boxed{xy leq 9}$ you get
                $$boxed{6 < xy leq 9}$$







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Nov 26 at 5:28

























                answered Nov 26 at 4:50









                trancelocation

                8,6421520




                8,6421520






























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