The combined equation of two lines L and L₁ is 2 x^2+a x y+3 y^2=0 and the combined equation of two lines L and L₂ is 2 x^2+b x y-3 y^2=0 . If L₁ and L₂ are perpendicular, then a^2+b^2=

Step-by-step solution

Let L y=m x, L₁ y=k x, L₂ y=- 1 k x Now, aligned Now, & (y-m x)(y-k x)=0 & y^2-y k x-m x y+m k x^2=0 & m k x^2-(k+m) x y+y^2=0 aligned Given that, 2 x^2+a x y+3 y^2=0 or 2 3 x^2+ a 3 x y+y^2=0 Now, (y-m x) (y+ x k )=0 array rlrl & & y^2+ x y k -m x y- m x^2 k & =0 & - m k x^2+ ( 1 k -m ) x y+y^2 & =0 array which is or aligned 2 x^2+b x y-3 y^2 & =0 - 2 3 x^2- b 3 x y+y^2 & =0 aligned On comparing, -m k = -2 3 ,- b 3 = 1 k -m By solving Eqs. (i) and (ii), we get aligned m & = 2 3 , k=1 and a=-5, b=-1 a^2+b^2 & =25+1