# Straight Line in Normal Form

We will learn how to find the equation of a straight line in normal form.

The equation of the straight line upon which the length of the perpendicular from the origin is p and this perpendicular makes an angle α with x-axis is x cos α + y sin α = p

If the line length of the perpendicular draw from the origin upon a line and the angle that the perpendicular makes with the positive direction of x-axis be given then to find the equation of the line.

Suppose the line AB intersects the x-axis at A and the y-axis at B. Now from the origin O draw OD perpendicular to AB.

The length of the perpendicular OD from the origin = p and ∠XOD = α, (0 ≤ α ≤ 2π).

Now we have to find the equation of the straight line AB.

Now, from the right-angled ∆ODA we get,

$$\frac{OD}{OA}$$ = cos α

$$\frac{p}{OA}$$ = cos α

OA = $$\frac{p}{cos α}$$

Again, from the right-angled ∆ODB we get,

∠OBD = $$\frac{π}{2}$$ - ∠BOD = ∠DOX = α

Therefore, $$\frac{OD}{OB}$$ = sin α

or, $$\frac{p}{OB}$$ = sin α

or, OB = $$\frac{p}{sin α}$$

Since the intercepts of the line AB on x-axis and y-axis are OA and OB respectively, hence the required

$$\frac{x}{OA}$$ + $$\frac{y}{OB}$$ = 1

$$\frac{x}{\frac{p}{cos α}}$$ + $$\frac{y}{\frac{p}{sin α}}$$ = 1

$$\frac{x cos α}{p}$$ + $$\frac{y sin α}{p}$$ = 1

x cos α + y sin α = p, which is the required form.

Solved examples to find the equation of a straight line in normal form:

Find the equation of the straight line which is at a of distance 7 units from the origin and the perpendicular from the origin to the line makes an angle 45° with the positive direction of x-axis.

Solution:

We know that the equation of the straight line upon which the length of the perpendicular from the origin is p and this perpendicular makes an angle α with x-axis is x cos α + y sin α = p.

Here p = 7 and α = 45°

Therefore, the equation of the straight line in normal form is

x cos 45° + y sin 45° = 7

⇒ x ∙ $$\frac{1}{√2}$$ + y ∙ $$\frac{1}{√2}$$ = 7

⇒ $$\frac{x}{√2}$$ + $$\frac{y}{√2}$$ = 7

⇒ x + y = 7√2, which is the required equation.

Note:

(i) The equation of a, straight line in the form of x cos α + y sin α = p is called its normal form.

(ii) In equation x cos α + y sin α = p, the value of p is always positive and 0 ≤ α≤ 360°.

The Straight Line