If θ1,θ2,θ3 are respectively the angles by which the coordinate axes are to be rotated to eliminate the xy term from the following equations, then the descending order of these angles is A1 = 3x2+5xy+3y2+2x+3y+4 = 0 a2 = 5x2+2√3xy+3y2+6 = 0 A3 = 4x2+√3xy+5y2−4 = 0
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