Question

Suppose ax² + 2hxy + by² = 0 represents a pair of lines and (x_1, y₁) is a point in the plane. Then

 1.  The equation of the pair of lines passing through (x₁, y₁) and parallel to these lines is a(x − x₁)² + 2h(x − x₁) (y − y₁) + b(y − y₁)² = 0. 

2.  The equation of the pair of lines passing through (x₁, y₁) and perpendicular to the given lines is b(x − x₁)² −  2h(x − x₁) (y − y₁) + a(y − y₁)² = 0.

Answer 1

Suppose the lines represented by ax² + 2hxy + by² = 0 are l₁x + m₁y = 0 and l₂x + m₂y = 0. Hence, l₁l₂ = a, l₁m₂ + l₂m₁ = 2h and m₁m₂ = b 

1.  the equations of the lines through (x₁, y₁) and parallel to the lines l₁x + m₁y = 0 and l₂x + m₂y = 0 are l₁ (x − x₁) + m₁ (y − y₁) = 0 and l₂ (x − x₁) + m₂ (y − y₁) = 0.

Hence, their combined equation is

2.  the equations of the lines through (x₁, y₁) and perpendicular to the lines are m₁ (x − x₁) −  l₁ (y − y₁) = 0 and m₂ (x − x₁) − l₂ (y − y₁) = 0.

Hence, their combined equation is