#### 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) `x – sqrt3y + 8 = 0`

(ii) *y *– 2 = 0

(iii) *x *– *y *= 4

#### Solution

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`sqrt(0^2 + 1^2) = 1`, 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