If the lines x^2+k x y+y^2=0 and x+y=1 make up the sides of an equilateral triangle, then the value of k^2 equals

Step-by-step solution

Lines x^2+k x y+y^2 and x+y-1=0 make the sides of equilateral triangle.(given) x^2+k x y+y^2=0 represent a pair of straight lines passing through origin. Let m be the slope of one line whose equation is y=m x According to the question, aligned & = | m₁-m₂ 1+m₁ m₂ |, = angle between lines & 60^ = | m-(-1) 1+m(-1) |[ slope of x+y=1 is -1] & 3 = m-1 1-m & (1-m)^2=(m+1)^2 & 2 m^2-8 m+2=0 & m^2-4 m+1=0 aligned aligned & ( y x )^2-4 ( y x )+1=0 [ y=m x y x =m ] & y^2 x^2 - 4 y x +1=0 y^2-4 x y+x^2=0 aligned Compare with