Giải HPT: `{(x^2 + y^2 + 1/x^2 + 1/y^2 = 5),((xy-1)^2 = x^2 – y^2 + 2):}`

1 bình luận về “Giải HPT: `{(x^2 + y^2 + 1/x^2 + 1/y^2 = 5),((xy-1)^2 = x^2 – y^2 + 2):}`”

1. {(x^2+y^2+1/(x^2)+1/(y^2)=5(1)),((xy-1)^2=x^2-y^2+2(2)):}(Đk:x,yne0)

    (2)<=>(xy-1)^2=x^2-y^2+2

   <=>x^2y^2-2xy+1=x^2-y^2+2

   <=>x^2y^2-x^2+y^2+1-2=2xy

   <=>x^2y^2-x^2+y^2-1=2xy

   <=>x^2(y^2-1)+(y^2-1)=2xy

   <=>(x^2+1)(y^2-1)=2xy

   <=>((x^2+1)(y^2-1))/(xy)=2

   <=>(x^2+1)/x . (y^2-1)/y=2

   <=>(x+1/x)(y-1/y)=2

   (1)<=>x^2+y^2+1/(x^2)+1/(y^2)=5

   <=>x^2+y^2+1/(x^2)+1/(y^2)+2-2=5

   <=>(x^2+2+1/(x^2))+(y^2-2+1/(y^2))=5

   <=>(x+1/x)^2+(y-1/y)^2=5

   Đặt {(a=x+1/x),(b=y-1/y):}

   Khi đó ta có hệ:

   {(a^2+b^2=5)(3),(ab=2):}

   (3)<=>a^2+b^2=5

   <=>a^2+2ab+b^2=5+2ab

   <=>(a+b)^2=5+2.2

   <=>(a+b)^2=9

   <=>[(a+b=3),(a+b=-3):}

   +)Với {(a+b=3),(ab=2):}

   <=>{(a=-2),(b=-1):}

   <=>{(x+1/x=-2),(y-1/y=-1):}

   <=>{(x=-1),(y=(-1+-sqrt5)/2):}

   +)Với {(a+b=-3),(ab=2):}

   <=>{(a=2),(b=-1):}

   <=>{(x+1/x=2),(y+1/y=-1):}

   <=>{(x=1),(y=(1+-sqrt5)/2):}

   Trả lời