If coordinate axes are rotated through an angle
θ in anti-clockwise direction, then
x=X‌cos‌θ−Y‌sin‌θ and
y=X‌sin‌θ+Y‌cos‌θ, on substituting the values of
x and
y in equation
x2+y2+2xy+2x+6y+1=0, we get
(X‌cos‌θ−Y‌sin‌θ)2+(X‌sin‌θ+Y‌cos‌θ)2+2(X‌cos‌θ−Y‌sin‌θ)(X‌sin‌θ+Y‌cos‌θ)
+2(X‌cos‌θ−Y‌sin‌θ)+6(X‌sin‌θ+Y‌cos‌θ)+1=0
⇒(cos2θ+sin2θ+2‌sin‌θ‌cos‌θ)X2+(−2‌sin‌θ‌cos‌θ+2‌sin‌θ‌cos‌θ+2cos2θ−2sin2θ)XY+(sin2θ+cos2θ−2‌sin‌θ‌cos‌θ)Y2
+(2‌cos‌θ+6‌sin‌θ)X+(6‌cos‌θ−2‌sin‌θ)Y+1=0
⇒(1+sin‌2‌θ)+2‌cos‌2‌θ‌X‌Y+(1−sin‌2‌θ)Y2+(2‌cos‌θ+6‌sin‌θ)X+(6‌cos‌θ−2‌sin‌θ)Y+1=0
On comparing with transformed given equation
(2+√3)X2+2XY+(2−√3)Y2+aX+bY+2=0
we get
and
b=4(3‌cos‌θ−sin‌θ)=6√3−2 ∴3a−b=6√3+18−6√3+2=20