If the equation x²-3 x y+ y²+3 x-5 y+2=0 represents a pair of lines, where is real number and is angle between them, then value of cosec ² is

Step-by-step solution

Comparing given equation with a x²+2 h x y+b y²+2 g x+2 f y+c=0 , we get a =1, ~h =- -3 2 , ~b = , g = 3 2 , f = -5 2 , c =2 Given equation represents pair of lines only if | array lll a & h & g h & b & f g & f & c array |=0 aligned & | array ccc 1 & - 3 2 & 3 2 3 & & - 5 2 array |=0 & | array ccc 3 2 & - 5 2 & 2 array | & 1 2 1 2 1 2 | array ccc 2 & -3 & 3 -3 & 2 & -5 3 & -5 & 4 array |=0 & 2(8 -25)+3(-12+15)+3(15-6 )=0 & 16 -50+9+45-18 =0 &-2 =-4 =2 b =2 aligned Now, aligned &= | 2 h²-a b a+b | &= | 2 9 4 -2 1+2