Find equation of the line through the point (0, 2) making an angle with the positive x-axis. Also, find the equation of line parallel to it and crossing the y-axis at a distance of 2 units below the origin.

Given point = (0, 2) and θ = 2π/3

We know that m = tanθ

∴ m = tan (2π/3) = -√3

We know that the point (x, y) lies on the line with slope m through the fixed point (x₀, y₀), if and only if, its coordinates satisfy the equation y – y₀ = m (x – x₀)

∴ y – 2 = -√3 (x – 0)

⇒ y – 2 = -√3 x

⇒ √3 x + y – 2 = 0

Given, equation of line parallel to above obtained equation crosses the y-axis at a distance of 2 units below the origin.

So, the point = (0, -2) and m = -√3

From point slope form equation,

⇒ y – (-2) = -√3 (x – 0)

⇒ y + 2 = -√3 x

⇒ √3 x + y + 2 = 0

Ans. The equation of line is √3 x + y – 2 = 0 and the line parallel to it is √3 x + y + 2 = 0.