Question

The acute angle between the lines $y=3$ and $\sqrt{3}x+9$ is:

• A. $30°$
• B. $60°$
• C. $45°$
• D. $90°$

Answer 1

The correct option is B

$60°$

Step 1: Finding the slope of both the given lines.

The given equations are $y=3$ and $\sqrt{3}x+9$

Slope of $y=3$:

$$\begin{gathered} y=3 \\ \Rightarrow y=0.x+3\left[\text{equation is of the form}y=mx+b\right] \\ \therefore m_1=0 \end{gathered}$$

Slope of $\sqrt{3}x+9$:

$$\begin{gathered} \sqrt{3}x+9=0 \\ \Rightarrow 0.y=\sqrt{3}.x+9\left[\text{equation is of the form}y=mx+b\right] \\ \therefore m_2=\sqrt{3} \end{gathered}$$

Step 2: Finding the angle between the two lines.

$$\begin{gathered} \tan \theta =\frac{(m_2-m_1)}{1+m_1.m_2}\times m_2\left[\theta =\text{angle between}{m}_2\text{and}{m}_1\right] \\ =\frac{\sqrt{3}-0}{1+0.\sqrt{3}} \\ =\sqrt{3} \\ \therefore \theta ={\tan }^{-1}\left(\sqrt{3}\right) \\ =60° \end{gathered}$$

Hence, the correct answer is Option (B).