Question

Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.

(i) (ii) y – 2 = 0 (iii) x – y = 4

Answer 1

(i) The given equation is.

It can be reduced as:

On dividing both sides by, we obtain

Equation (1) is in the normal form.

On comparing equation (1) with the normal form of equation of line

x cos ω + y sin ω = p, we obtain ω = 120° and p = 4.

Thus, the perpendicular distance of the line from the origin is 4, while the angle between the perpendicular and the positive x-axis is 120°.

(ii) The given equation is y – 2 = 0.

It can be reduced as 0.x + 1.y = 2

On dividing both sides by, we obtain 0.x + 1.y = 2

⇒ x cos 90° + y sin 90° = 2 … (1)

Equation (1) is in the normal form.

On comparing equation (1) with the normal form of equation of line

x cos ω + y sin ω = p, we obtain ω = 90° and p = 2.

Thus, the perpendicular distance of the line from the origin is 2, while the angle between the perpendicular and the positive x-axis is 90°.

(iii) The given equation is x – y = 4.

It can be reduced as 1.x + (–1) y = 4

On dividing both sides by, we obtain

Equation (1) is in the normal form.

On comparing equation (1) with the normal form of equation of line

x cos ω + y sin ω = p, we obtain ω = 315° and .

Thus, the perpendicular distance of the line from the origin is, while the angle between the perpendicular and the positive x-axis is 315°.