Question

10. Obtain the equation of the straight line in the
normal form, for a (the angle between the
perpendicular to the line from the origin and
the x-axis) for each of the following, on the
same graph:
(i)
a < 90°
(ii) 90° <a< 180°
(iii) 180º < a < 270°
(iv) 270º < a < 360°​

Answer 1

Answer:

Step-by-step explanation:

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.

Straight Line in Normal Form

Straight Line in Normal Form

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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,

ODOA = cos α        

⇒ pOA = cos α          

⇒ OA = pcosα

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

∠OBD = π2 - ∠BOD = ∠DOX = α    

Therefore, ODOB = sin α

or, pOB = sin α      

or, OB = psinα

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

xOA + yOB = 1        

⇒ xpcosα + ypsinα = 1

⇒ xcosαp + ysinαp = 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 ∙ 1√2 + y ∙ 1√2 = 7

⇒ x√2 + 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°.