Question

What is the acute angle between the two straight lines y = ( ) 2– 3 x + 5 and y = ( ) 2 3 + x – 7 ?

(a) 60° 

(b) 45° 

(c) 30° 

(d) 15°

Answer 1

(a) 60º

The two lines are: y = \((2-\sqrt3)\) \(x\) + 5            ...(i) 

y = \((2-\sqrt3)\) \(x\) – 7               ...(ii) 

Slope of line (i), m₁ = \(2-\sqrt3\)

Slope of line (ii), m₂ = \(2+\sqrt3\)

If θ is the angle between the two lines, then 

tan θ = \(\big|\frac{m_1-m_2}{1+m_1m_2}\big|\) = \(\bigg|\frac{(2-\sqrt3)-(2+\sqrt3)}{1+(2-\sqrt3)(2+\sqrt3)}\bigg|\)

= \(\bigg|\frac{-2\sqrt3}{1+1}\bigg|=\sqrt3\)

∴ θ = –1 tan ( 3) = 60º.