Suppose the axes are to be rotated through an angle so as to remove the x y form from the equation 3 x^2+2 3 x y+y^2=0 . Then in the new coordinate system the equation x^2+y^2+2 x y=2 is transformed to

Step-by-step solution

Let axes be rotated through an angle , then old coordinates are aligned & x=X -Y & y=X +Y aligned Putting given equation 3 x^2+2 3 x y+y^2=0 aligned & 3(X -Y )^2+2 3 (X -Y ) & (X +Y )+(X +Y )^2=0 aligned aligned & 3 ^2 X^2+3 ^2 Y^2-6 X Y & +2 3 (X^2 +X Y ^2 -X Y ^2 . & .-Y^2 )+X^2 ^2 +Y^2 ^2 & +2 X Y =0 aligned aligned & (3 ^2 0+2 3 + ^2 ) X^2 &+ (3 ^2 -2 3 + ^2 ) Y^2 &+ (-6 +2 3 ^2 -2 3 ^2 . &+2 ) X Y=0 aligned To remove X Y term -3 2 +2 3 2 + 2 =0 2 = 3 =30^ New transformed equation of x^2+y^2+2 x y=2 aligned ali