Question

A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of 60° with the line x + y = 0. Then an equation of the line L is :
(A) (√3 - 1)x +(√3 + 1)y = 8√2 (B) √(3)x + y = 8 (C) x + √(3)y = 8 (D) (√3 + 1)x +(√3 - 1)y = 8√2

Answer 1

Answer:

c verify experimentally

Answer 2

Equation of line is

x (√3+1) + y (√3-1) = 8√2

option D is correct

•Length of perpendicular from

origin is 4 units

i.e. p = 4

•Now , Angle between perpendicular and x + y = 0 is 60°

•slope of x + y = 0 is -1

m = -1

•let the slope of perpendicular is M

• Also ,

tan60° = (M -m )/1+Mm

√3 =( M +1 )/ 1-M

√3 - √3M = M + 1

√3 -1 = M(√3 +1)

(√3-1)/(√3+1) = M

•on Rationalizing the denominator

M = (3+1-2√3)/(3-1)

M = (4-2√3)/2

M = 2-√3

=> angle between origin and perpendicular is tan^-1(2-√3)

i.e. A = π/12

•sinA = (√3-1)/2√2

•CosA = (√3+1)/2√2

•Now using Normal form of line

xcosA + ysinA = p

x (√3+1)/2√2 + y (√3-1)/2√2 = 4

x (√3+1) + y (√3-1) = 8√2

•Equation of line is

x (√3+1) + y (√3-1) = 8√2