Question

Question 3 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 -- 3^0.5 y + 8 = 0

(ii) y – 2 = 0

(iii) x – y = 4

Class X1 - Maths -Straight Lines Page 227

Answer 1

if any equation of the form ax + by + c = 0 can be converted into normal form (xcosα+ysinα = P) dividing each term by √(a² + b²)

(i) x - √3y + 8 = 0
dividing √(1² + √3²) = √(1+3) = 2
x/2 - √3y/2 + 8/2 = 0
(1/2)x + (-√3/2)y + 4 = 0
(-1/2)x + (√3/2)y = 4

if we put, cosα = - 1/2
cos(180° - 60°) = -1/2
cos120° = -1/2
hence, α = 120° , and P = 4
now, equation is x.cos120° + ysin120° = 4


(ii) y - 2 = 0
0.x + y = 2
dividing √(0²+1²) = 1
0.x/1 + y/1 = 2/1
(0).x + (1)y = 2
put cosα = 0 = cos90°
hence, α = 90° , P = 2
now, equation is x.cos90° + y.sin90° = 2


(iii) x - y = 4
dividing √{1²+(-1)²} = √2
x/√2 - y/√2 = 4/√2
(1/√2)x + (-1/√2)y = 2√2
put sinα = -1/√2 = sin(360°-45°)
[ use the formula, sin(360°-∅) = -sin∅]
sin315° = sinα,
hence, α = 315° and P = 2√2
now, equation is x.cos315° + y.sin315° = 2√2

Answer 2

Answer:

I hope this is the correct answer