If the lines represented by a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y + c = 0 intersect on the x -axis, which of the following is in general incorrect

Step-by-step solution

Given, a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y + c = 0 put y = 0 ⇒ a x 2 + 2 g x + c = 0 Now roots will be equal to given axis so, D = 0 ⇒ g 2 = a c   . . . . . . 1 Now for pair of straight line a b c + 2 f g h - a f 2 - b g 2 - c h 2 = 0 Since g 2 = a c So 2 f g h = a f 2 + c h 2   . . . . . . . 2 Also 2 f g h 2 = a f 2 + c h 2 2 ⇒ 4 f 2 g 2 h 2 = a f 2 + c h 2 2 ⇒ 4 f 2 a c h 2 = a f 2 + c h 2 2 ⇒ a f 2 + c h 2 2 - 4 f 2 a c h 2 = 0 ⇒ a f 2 - c h 2 2 = 0 ⇒ a f 2