Given, equation is 3x2+3y2+2xy=2 ...(i) When coordinate axis are rotated through an angle θ , then x=X‌cos‌θ−Y‌sin‌θ and y=X‌sin‌θ+Y‌cos‌θ where, (X,Y) is new coordinate after transformation. Given, θ=45° , then x=
x
√2
−
y
√2
and y=
x
√2
+
y
√2
Putting value of x and y in Eq. (i), we get 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+(X2−Y2)=2 ⇒
3
2
[2X2+2Y2]+X2−Y2=2 ⇒3X2+3Y2+X2−Y2=2 ⇒4X2+2Y2=2 or 2X2+Y2=1 ∴ Transformed equation is 2x2+y2=1