If the pair of lines 3 x^2-5 x y+p y^2=0 and 6 x^2-x y-5 y^2=0 have one line common, then p =

Step-by-step solution

Let the two pairs of lines be given by: 3x^2 - 5xy + py^2 = 0 6x^2 - xy - 5y^2 = 0 Assume the common line has slope m , so its equation is y = mx . Substitute into the second pair's equation and divide by x^2 : 6 - m - 5m^2 = 0 5m^2 + m - 6 = 0 Factorizing: (m - 1)(5m + 6) = 0 m = 1 or m = - 6 5 Substitute each slope into the first pair's equation 3 - 5m + pm^2 = 0 : For m = 1 : 3 - 5(1) + p(1)^2 = 0 p = 2 For m = - 6 5 : 3 - 5 (- 6 5 ) + p ( 36 25 ) = 0 9 + 36p 25 = 0 p = - 25 4 Thus, p can be either 2 or - 25 4 .