Question

The equation of the straight line passing through (x₁, y₁) and making an angle P with the positive direction of the x-axis measured in counter clock sense is

(x - x₁) : cos θ = (y - y₁) : sin θ

Answer 1

Suppose the line is L which passes through (x₁, y₁).

Case 1: The line L is vertical so that θ =  π/2. Since the  equation of L is x = x₁ which is written as (x - x₁) : cosθ =(y - y₁) : sin θ because cos θ = 0 and sinθ  = 1.

Case 2: Suppose L is non-vertical so that θ ≠ π/ 2. Hence , its equation is

which shows that |γ| represents the distance of the point (x, y) on the line from the given point (x₁, y₁). Therefore, if (x, y) is any point on the line and γ is any real parameter, then the locus of the point (x₁ +  γcosθ, y₁ +  γ sinθ) is the straight line

(x - x₁) : cosθ = (y - y₁): sinθ

 Also, γ takes positive values for points on the line on one side of (x₁, y₁) and takes negative values for the points on the other side of (x₁, y₁). For a given positive value of γ , there will be two points on the line which are equidistant from (x₁, y₁). Further, the equations

x = x₁ +  γcosθ and y = y₁ +  γsinθ

(θ is a fixed and γ is a parameter) are called the parametric equations of the line passing through (x₁, y₁) and making angle θ with the positive direction of the x-axis measured in counter clock sense.