Reduce the following equations into normal form. Find their perpendicular distance from the origin and angle between perpendicular and the positive x-axis
(i) x - √3 y + 8 = 0    (ii) y - 2 = 0    (iii) x – y = 4

Solution:

(i) The given equation is x - √3 y + 8 = 0

It can be written as

x - √3 y = - 8

- x + √3 y = 8 (because the right side number in normal form should be non-negative)

On dividing both sides by √(- 1)² + (√3)² = √4 = 2 we obtain

- x/2 + √3/2y = 8/2

(- 1/2)x + (√3/2)y = 4

x cos 120° + y sin 120° = 4 ....(1)

Equation (1) is in the normal form

On comparing equation (1) with the normal form of the equation of the 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 represented as 0.x + 1.y = 2

On dividing both sides by √0² + 1² = 1 we obtain

0.x + 1.y = 2

⇒ x cos 90° + y sin 90° = 2 ....(2)

Equation (2) is in the normal form.

On comparing equation (2) with the normal form of the equation of a 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 √[(1)² + (- 1)²] = √2 we obtain

(1/√2)x + (- 1/√2)y = 4/√2

⇒ x cos (2π - π/4) + y sin (2π - π/4) = 2

⇒ x cos 315° + y sin 315° = 2√2 ....(3)

Equation (3) is in the normal form.

On comparing equation (3) with the normal form of the equation of the line

x cos ω + y sin ω = p , we obtain ω = 315° and p = 2√2

Thus, the perpendicular distance of the line from the origin is 2√2 while the angle between the perpendicular and the positive x-axis is 315°

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NCERT Solutions Class 11 Maths Chapter 10 Exercise 10.3 Question 3

Reduce the following equations into normal form. Find their perpendicular distance from the origin and angle between perpendicular and the positive x-axis. (i) x - 3√y + 8 = 0 (ii) y - 2 = 0 (iii) x – y = 4

Summary:

i) x cos 120° + y sin 120° = 4, where 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) x cos 90° + y sin 90° = 2, where 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) x cos 315° + y sin 315° = 2√2, where the perpendicular distance of the line from the origin is 2√2 while the angle between the perpendicular and the positive x-axis is 315°