Lines to be Coincident, Perpendicular & Equation of Second Degree, Bisectors

Condition for the Lines to be Coincident:

The lines are coincident if the angle between them is zero. Thus, lines are coincident.

ó θ = 0

ó tanθ = 0

ó 2√ (h² - ab)/ a + b = 0

ó h² – ab = 0

ó h² = ab

Hence, the lines represented by ax² + 2hxy + by² = 0 are coincident if h² = ab.

Condition for the Lines to be Perpendicular:

The lines are perpendicular if the angle between them is π/2.

Thus, lines are perpendicular.

ó θ = π/2

ó cot θ = cot π/2

ó cot θ = 0

ó a + b/ 2√ (h² - ab) = 0

ó a + b = 0

ó coeff of x² + coeff of y² = 0

Thus, the lines represented by  are perpendicular if a + b = 0 i.e., coeff. of x² + coeff, of y² = 0

General Equation of Second Degree:

The necessary and sufficient condition for ax² + 2hxy + by² + 2gx + 2fy + c = 0 to represent a pair of straight lines is that abc + 2fgh – af² – bg² – ch² = 0 or,

Equations of the Bisectors:

The equations of the bisectors of the angles between the lines (L₁, L₂) represented by ax² + 2hxy + by² + 2gx + 2fy + c = 0 are given by [(x - x₁)² - (y - y₁)²]/(a - b) = [(x - x₁) - (y - y₁)]/h, where (x₁, y₁) the point of intersection of the lines is represented by the given equation.