Question

The obtuse angle between the lines $x-\sqrt{3}y=5$ and $\sqrt{3}x-y=7$ is

• A. ${105}^{\circ }$
• B. ${150}^{\circ }$
• C. ${120}^{\circ }$

Answer 1

The correct option is B ${150}^{\circ }$

The given lines are:
$x-\sqrt{3}y=5$ and $\sqrt{3}x-y=7$

We can write them as:
$y=\frac{1}{\sqrt{3}}-\frac{5}{\sqrt{3}}$ and $y=\sqrt{3}x-7$

Here, $m_1=\frac{1}{\sqrt{3}}$ and $m_2=\sqrt{3}$

If the angle between the lines is $\theta$, then

$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

Here, we get $\tan \theta =\left|\frac{\sqrt{3}-\frac{1}{\sqrt{3}}}{1+\sqrt{3}\times \frac{1}{\sqrt{3}}}\right|$

$\Rightarrow \tan \theta =\frac{2}{\sqrt{3}\times 2}$

$\Rightarrow \tan \theta =\frac{1}{\sqrt{3}}$

Thus, $\theta ={30}^{\circ }$. That's the acute angle.

The obtuse angle between the two lines is ${180}^{\circ }-{30}^{\circ }={150}^{\circ }$