Condition on $lambda$ so that the equation is satisfied












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Let there exist real numbers $a,b,c,d$ such that $lambda x^2+2xy+y^2=(ax+by)^2+(cx+dy)^2 forall x,yinmathbb{R}$, then what is the condition on $lambda$?



I think such a $lambda$ does not exist. I got the equations $$begin{aligned} ab+cd&=1\ a^2+c^2&=lambda \ b^2+d^2&=1end{aligned}$$ but am unable to proceed further. Any hints? Thanks beforehand.






algebra-precalculus polynomials







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    $begingroup$


    Let there exist real numbers $a,b,c,d$ such that $lambda x^2+2xy+y^2=(ax+by)^2+(cx+dy)^2 forall x,yinmathbb{R}$, then what is the condition on $lambda$?



    I think such a $lambda$ does not exist. I got the equations $$begin{aligned} ab+cd&=1\ a^2+c^2&=lambda \ b^2+d^2&=1end{aligned}$$ but am unable to proceed further. Any hints? Thanks beforehand.






    algebra-precalculus polynomials







    share|cite|improve this question











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      0








      0





      $begingroup$


      Let there exist real numbers $a,b,c,d$ such that $lambda x^2+2xy+y^2=(ax+by)^2+(cx+dy)^2 forall x,yinmathbb{R}$, then what is the condition on $lambda$?



      I think such a $lambda$ does not exist. I got the equations $$begin{aligned} ab+cd&=1\ a^2+c^2&=lambda \ b^2+d^2&=1end{aligned}$$ but am unable to proceed further. Any hints? Thanks beforehand.






      algebra-precalculus polynomials







      share|cite|improve this question











      $endgroup$




      Let there exist real numbers $a,b,c,d$ such that $lambda x^2+2xy+y^2=(ax+by)^2+(cx+dy)^2 forall x,yinmathbb{R}$, then what is the condition on $lambda$?



      I think such a $lambda$ does not exist. I got the equations $$begin{aligned} ab+cd&=1\ a^2+c^2&=lambda \ b^2+d^2&=1end{aligned}$$ but am unable to proceed further. Any hints? Thanks beforehand.






      algebra-precalculus polynomials




      algebra-precalculus polynomials






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      share|cite|improve this question













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      edited Dec 12 '18 at 11:14







      vidyarthi

















      asked Dec 12 '18 at 10:55









      vidyarthividyarthi

      2,9641832




      2,9641832






















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          $begingroup$

          Hint the right-hand side is never negative






          share|cite|improve this answer









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          • $begingroup$
            I think it should be $ax+by$ and $cx+dy$
            $endgroup$
            – Empy2
            Dec 12 '18 at 11:11










          • $begingroup$
            yes, thanks, edited.
            $endgroup$
            – vidyarthi
            Dec 12 '18 at 11:15











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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          Hint the right-hand side is never negative






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I think it should be $ax+by$ and $cx+dy$
            $endgroup$
            – Empy2
            Dec 12 '18 at 11:11










          • $begingroup$
            yes, thanks, edited.
            $endgroup$
            – vidyarthi
            Dec 12 '18 at 11:15
















          2












          $begingroup$

          Hint the right-hand side is never negative






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I think it should be $ax+by$ and $cx+dy$
            $endgroup$
            – Empy2
            Dec 12 '18 at 11:11










          • $begingroup$
            yes, thanks, edited.
            $endgroup$
            – vidyarthi
            Dec 12 '18 at 11:15














          2












          2








          2





          $begingroup$

          Hint the right-hand side is never negative






          share|cite|improve this answer









          $endgroup$



          Hint the right-hand side is never negative







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 12 '18 at 11:03









          Empy2Empy2

          33.5k12261




          33.5k12261












          • $begingroup$
            I think it should be $ax+by$ and $cx+dy$
            $endgroup$
            – Empy2
            Dec 12 '18 at 11:11










          • $begingroup$
            yes, thanks, edited.
            $endgroup$
            – vidyarthi
            Dec 12 '18 at 11:15


















          • $begingroup$
            I think it should be $ax+by$ and $cx+dy$
            $endgroup$
            – Empy2
            Dec 12 '18 at 11:11










          • $begingroup$
            yes, thanks, edited.
            $endgroup$
            – vidyarthi
            Dec 12 '18 at 11:15
















          $begingroup$
          I think it should be $ax+by$ and $cx+dy$
          $endgroup$
          – Empy2
          Dec 12 '18 at 11:11




          $begingroup$
          I think it should be $ax+by$ and $cx+dy$
          $endgroup$
          – Empy2
          Dec 12 '18 at 11:11












          $begingroup$
          yes, thanks, edited.
          $endgroup$
          – vidyarthi
          Dec 12 '18 at 11:15




          $begingroup$
          yes, thanks, edited.
          $endgroup$
          – vidyarthi
          Dec 12 '18 at 11:15


















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