If the equation a x^2+2 h x y+b y^2+2 g x+2 f y+c=0 represents two straight lines equidistant from the origin, then f^4-g^4=

Step-by-step solution

Let the lines represented by equation aligned & a x^2+2 h x y+b y^2+2 g x+2 f y+c=0 & are y=m₁ x+c₁ and y=m₂ x+c₂ & aligned & So, a x^2+2 h x y+b y^2+2 g x+2 f y+c &= (m₁ x-y+c₁ ) (m₁ x-y+c₂ ) & m₁ m₂ a = - (m₁+m₂ ) 2 h = 1 b &= m₁ C₂+m₂ C₁ 2 g = - (C₁+C₂ ) 2 f = C₁ C₂ c aligned aligned Now, according to the information given in question, as lines are equidistance from origin, so aligned & |C₁ | 1+m₁^2 = |C₂ | 1+m₂^2 & c₁^2 (1+m₂^2 )=c₂^2 (1+m₁^2 ) & c₁^2-c₂^2=c₂^2 m₁^2-c₁^2 m₂^2 & (C₁+C₂ ) (C₁-C₂ )= (C₂ m₁+C₁ m₂ )