**Condition for the lines to be coincident: **The lines are coincident if the angle between them is zero. Thus, lines are coincident.

⇒ θ = 0

⇒ tanθ = 0

⇒ \(\frac{2\sqrt{{{h}^{2}}-ab}}{a+b}=0\).

⇒ h^{2} – ab = 0

⇒ h^{2} = ab

Hence, the lines represented by ax^{2} + 2hxy + by^{2} = 0 are coincident if h^{2} = ab.

**Condition for the lines to be perpendicular: **The lines are perpendicular if the angle between them is π/2.

Thus, lines are perpendicular.

⇒ θ = π/2

⇒ cot θ = cot π/2

⇒ cot θ = 0

⇒ \(\frac{a+b}{2\sqrt{{{h}^{2}}-ab}}=0\).

⇒ a + b = 0

⇒ coeff of x² + coeff of y² = 0

Thus, the lines represented by ax² + 2hxy + by² = 0 are perpendicular if a + b = 0 i.e., coeff. of x^{2} + coeff. of y^{2} = 0

**GENERAL EQUATION OF SECOND DEGREE: **The necessary and sufficient condition for ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 to represent a pair of straight lines is that abc + 2fgh – af^{2} – bg^{2} – ch^{2} = 0 or, \(\left| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \\\end{matrix} \right|=0\).

**EQUATIONS OF THE BISECTORS: **The equations of the bisectors of the angles between the lines (L₁, L₂) represented by ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 are given by \(\frac{{{\left( x-{{x}_{1}} \right)}^{2}}{{\left( y-{{y}_{1}} \right)}^{2}}}{a-b}=\frac{\left( x-{{x}_{1}} \right)\left( y-{{y}_{1}} \right)}{h}\), where (x_{1}, y_{1}) the point of intersection of the lines is represented by the given equation.