Find the angle between the lines whose slopes are (2 − \(\sqrt{3}\))

1.	Find the angle between the lines whose slopes are (2 − \(\sqrt{3}\)) and (2 + \(\sqrt{3}\)).
Answer» Let the slopes of the two lines are m₁ = 2 − \(\sqrt{3}\) and m₂ = (2 + \(\sqrt{3}\)). Then, angle ‘θ’ between the lines is given by θ = tan⁻¹\(\frac{m_2 − m_1}{1 + m_1m_2}\) = tan⁻¹\(\frac{(2 + \sqrt{3}) − (2 − \sqrt{3})}{1 + (2 − \sqrt{3})(2 + \sqrt{3})}\) = tan⁻¹\(\frac{2\sqrt{3}}{1 + 4 − 3}\) = tan⁻¹(\(\sqrt{3}\)) = tan^{– 1}(tan 60°) = 60°

Find the angle between the lines whose slopes are (2 − \(\sqrt{3}\)) and (2 + \(\sqrt{3}\)).

Answer»

Let the slopes of the two lines are m₁ = 2 − \(\sqrt{3}\) and m₂ = (2 + \(\sqrt{3}\)).

Then, angle ‘θ’ between the lines is given by

θ = tan⁻¹\(\frac{m_2 − m_1}{1 + m_1m_2}\)

= tan⁻¹\(\frac{(2 + \sqrt{3}) − (2 − \sqrt{3})}{1 + (2 − \sqrt{3})(2 + \sqrt{3})}\)

= tan⁻¹\(\frac{2\sqrt{3}}{1 + 4 − 3}\)

= tan⁻¹(\(\sqrt{3}\))

= tan^{– 1}(tan 60°)

= 60°