Given line equation of 2-order: 3x+2=0 This equation looks like: a11x2+2a12xy+2a13x+a22y2+2a23y+a33=0 where a11=0 a12=0 a13=23 a22=0 a23=0 a33=2 To calculate the determinant Δ=a11a12a12a22 or, substitute Δ=0000 Δ=0 Because Δ is equal to 0, then Given equation is straight line - reduced to canonical form The center of the canonical coordinate system in OXY x0=x~cos(ϕ)−y~sin(ϕ) y0=x~sin(ϕ)+y~cos(ϕ) x0=0⋅0 y0=0⋅0 x0=0 y0=0 The center of canonical coordinate system at point O
(0, 0)
Basis of the canonical coordinate system e1=(1,0) e2=(0,1)
Invariants method
Given line equation of 2-order: 3x+2=0 This equation looks like: a11x2+2a12xy+2a13x+a22y2+2a23y+a33=0 where a11=0 a12=0 a13=23 a22=0 a23=0 a33=2 The invariants of the equation when converting coordinates are determinants: I1=a11+a22
I1=0 I2=0 I3=0 I(λ)=λ2 K2=−49 Because I2=0∧I3=0∧(I1=0∨K2=0) then by line type: this equation is of type : two coincident straight lines I1y~2+I1K2=0 or