The transformed equation 3 x^2+3 y^2+2 x y=2 , when the coordinate axes are rotated through an angle 45^ is

Step-by-step solution

Given, equation is 3 x^2+3 y^2+2 x y=2 (i) When coordinate axis are rotated through an angle , then x=X -Y and y=X +Y where, (X, Y) is new coordinate after transformation. Given, =45^ , then x= x 2 - y 2 and y= x 2 + y 2 Putting value of x and y in Eq. (i), we get aligned & 3 ( X 2 - Y 2 )^2+3 ( X 2 + Y 2 )^2 & +2 ( X 2 - Y 2 ) ( X 2 + Y 2 )=2 & 3 2 (X-Y)^2+ 3 2 (X+Y)^2+ (X^2-Y^2 )=2 & 3 2 [2 X^2+2 Y^2 ]+X^2-Y^2=2 & 3 X^2+3 Y^2+X^2-Y^2=2 & 4 X^2+2 Y^2=2 & or & 2 X^2+Y^2=1 & aligned Transformed equation is 2 x^2+y^2