The equation x^2-3 x y+ y^2+3 x-5 y+2=0 , where is real number, represents pair of lines. If is acute angle between the lines, then cosec ^2 10 =

Step-by-step solution

Using the condition for a pair of straight lines , the given equation satisfies abc + 2fgh - af^2 - bg^2 - ch^2 = 0 . For x^2 - 3xy + y^2 + 3x - 5y + 2 = 0 , the coefficients are a=1 , h=- 3 2 , g= 3 2 , b= , f=- 5 2 , c=2 . Substituting into the condition: 2 + 2 ( -5 2 ) ( 3 2 ) ( -3 2 ) - 1 ( 25 4 ) - ( 9 4 ) - 2 ( 9 4 ) = 0 2 + 45 4 - 25 4 - 9 4 - 18 4 = 0 2 - 9 4 + 2 4 = 0 - + 2 4 = 0 = 2 For the acute angle between the lines, = | 2 h^2 - ab a + b | . With a=1 , b=2 , h=- 3 2 : h^2 - ab = 9 4 - 2 = 1 4 = | 2 1