Question

If p be the length of the perpendicular from the origin on the straight line ax + by = p and b = \(\frac{\sqrt3}{2},\) then what is the angle between the perpendicular and the positive direction of x ? 

(a) 30º 

(b) 45º 

(c) 60º 

(d) 90º

Answer 1

(c) 60º

Since p is the length of perpendicular from origin on the straight line ax + by – p = 0.

p = \(\frac{|a.0+b.0-p|}{\sqrt{a^2+b^2}}\)

⇒ 1 = \(\sqrt{a^2+b^2}\) ⇒ 1 = a² + \(\frac{3}{4}\)             \(\big(\because{b} = \frac{\sqrt3}{2}\big)\)

⇒ a² = \(\frac{1}{4}\) ⇒ a = \(\frac{1}{2}.\)

∴ Equation of the straight line is \(\frac{1}{2}x\) + \(\frac{\sqrt3}{2}y\) = p

⇒ \(x\) cos 60° + y sin 60° = p 

Hence required angle is 60°, which is the angle between the perpendicular and the positive direction of x-axis.