Show that the lines x 2 - 4 x y + y 2 = 0 and x + y = 6 form an equilateral triangle. Find its area and perimeter.

Equation of the pair of lines is x 2 - 4 x y + y 2 = 0 -----(i) The point of intersection of the above lines is O 0 , 0 Let θ be the angle between the lines then ∴ tan θ = 2 h 2 - a b a + b ∴ tan θ = 2 - 2 2 - 1 × 1 1 + 1 = 2 4 - 1 2 ∴ tan θ = 3 ∴ θ = tan - 1 3 ∴ θ = 60 ° Let O A or O B has slope m, ∴ Its equation is y = m x Now the slope of x + y = 6 is - 1 Let us find the lines making 60 ∘ angle with the above line. ∴ tan 60 ∘ = m - ( − 1 ) 1 + m ( − 1 ) ∴ 3 = m + 1 1 − m Squaring both sides, we get ∴ 3 = ( m + 1 ) 2 ( 1 − m ) 2 ∴ 3 ( 1 − 2 m + m 2 ) = m 2 + 2 m + 1 ∴ 3 − 6 m + 3 m 2 = m 2 + 2 m + 1 ∴ 2 m 2 − 8 m + 2 = 0 ∴ m 2 − 4 m + 1 = 0 ∴ y x − 4 y x + 1 = 0 ∴ y 2 − 4 x y + x 2 = 0 ∴ x 2 − 4 x y + y 2 = 0 is the joint equation of two lines through the origin each making an angle of 60 ∘ with x + y = 6 ∴ x 2 − 4 x y + y 2 = 0 and x + y = 6 form a triangle O A B which is equilateral Let P = the length of the perpendicular from O to line x + y = 6 is ∴ c a 2 + b 2 ∴ 6 1 + 1 = 3 ∴ Area of ΔO A B = ( P ) 2 3 = ( 3 ) 2 3 = 3 3 = 3 sq units In right angled-triangle O A P ∴ sin 60 ∘ = O P O A ∴ 3 2 = 3 O A ∴ O A = 2 ∴ Length of each side of an equilateral triangle O A B = 2 units ∴ Perimeter of ΔOAB = 3 × length of each side = 3 × 2 = 6 units