Question

What is the acute angle between the lines represented by the equations $y-\sqrt{3}x-5=0$ and $\sqrt{3}y-x+6=0?$

• (A) ${30}^{\circ }$
• (B) ${45}^{\circ }$
• (C) ${60}^{\circ }$
• (D) ${75}^{\circ }$

Answer 1

The Correct Option is A

Explanation:
Given,$y-\sqrt{3}x-5=0$$\sqrt{3}y-x+6=0$$y-\sqrt{3}x-5=0$$\Rightarrow y=\sqrt{3x}+5$So, slope of line, $m_1=\sqrt{3}$$\sqrt{3}y-x+6=0$$\Rightarrow y=\frac{x}{\sqrt{3}}-\frac{6}{\sqrt{3}}$So, slope of the line, $m_2=\frac{1}{\sqrt{3}}$Let $\theta$ be the acute angle between the lines.The angle $\theta$ between the lines having slope $m_1$ and $m_2$ is given by:$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$$\Rightarrow \tan \theta =\left|\frac{\sqrt{3}-\frac{1}{\sqrt{3}}}{1+\sqrt{3}\times \frac{1}{\sqrt{3}}}\right|$$\Rightarrow \tan \theta =\left|\frac{\frac{2}{\sqrt{3}}}{2}\right|$$\Rightarrow \tan \theta =\frac{1}{\sqrt{3}}$$\Rightarrow \theta ={30}^{\circ }$Hence, the correct option is (A).