If x,y∈R  and 2x2+6xy+5y2=1,  then

Given equation is 2x2+6xy+5y2=1 -----(1)      Eq. (1) can be rewritten as 2x2+(6y)x+5y2−1=0               Since x  is real,            36y2−8(5y2−1)≥0 (1)        ⇒   y2≤2             ⇒  −2≤y≤2            Eq (1) can also be re written as            5y2+(6x)y+2x2−1=0 (2)      Sincey  is real,         36x2−20(2x2−1)≥0          ⇒       36x2−40x2+20≥0           ⇒   −4x2≥−20           ⇒    x2≤5          ⇒    −5≤x≤5