A1=3x2+5xy+3y2+2x+3 since, axes are rotated by an angle θ x=x1‌cos‌θ1−y1‌sin‌θ1 y=x1‌sin‌θ1+y1‌cos‌θ1 Substitute the value of x1 and y1 in A1 3(x1‌cos‌θ1−y1‌sin‌θ1)2+5(x1‌cos‌θ1−y1‌sin‌θ1) (x1‌sin‌θ1+y1‌cos‌θ1)+3(x1‌sin‌θ1+y1‌cos‌θ1)2 +2(x1‌cos‌θ1−y1‌sin‌θ1)+3(x1‌sin‌θ1+y1‌cos‌θ1)+4=0 Equatethe coefficient of x1 and y1 to zero. −6‌sin‌θ1‌cos‌θ1+5(cos‌θ12−sin‌θ12)+6‌sin‌θ1‌cos‌θ1=0 5‌cos‌2‌θ1‌‌=0 θ1‌‌=45∘ Similarly substitute x1 and y1 forA2 5(x1‌cos‌θ2−y1‌sin‌θ2)2+2√3(x1‌cos‌θ2−y1‌sin‌θ2)+ (x1‌sin‌θ2+y1‌cos‌θ2)+3(x1‌sin‌θ2+y1‌cos‌θ2)2+6=0 Set the coefficient of x1 and y1 to zero. −10‌sin‌θ2‌cos‌θ2+2√3(cos2θ2−sin2θ2)+6‌sin‌θ2‌cos‌θ2‌‌=0 −2‌sin‌2‌θ2+2√3‌cos‌θ2‌‌=0 tan‌2‌θ2‌‌=‌
2√3
2
θ2‌‌=30∘ Similarly for curve A3 −8‌sin‌θ3‌cos‌θ3+√3(cos2θ3−sin2θ3)+10‌sin‌θ3‌cos‌θ3‌‌=0 tan‌2‌θ3‌‌=‌
−√3
9
θ3‌‌=60∘ Therefore, the correct order is θ3>θ1>θ2