Question

The angle between the lines $y-x+5=0$ and $\sqrt{3}x-y+7=0$ is/are:

• A. ${15}^o$
• B. ${60}^o$
• C. ${165}^o$
• D. ${75}^o$

Answer 1

The correct option is A ${15}^o$

Equation of 1st line

$y-x+5=$

$y=x-5$

So, Slope $m_1=1$

Equation of 2nd line

$\sqrt{3}x-y+7$

$y=\sqrt{3}x+7$

So, slope $m_2=\sqrt{3}$

Now, angle between the two lines with slope $m_1$ and $m_2$ is

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

$=\frac{1-\sqrt{3}}{1+\sqrt{3}}$

$=\frac{1-\sqrt{3}}{1+\sqrt{3}}\times \frac{1-\sqrt{3}}{1-\sqrt{3}}$

$=\frac{(1-\sqrt{3}{)}^2}{1-3}$

$=\frac{1+3-2\sqrt{3}}{-2}$

$=\frac{2(2-\sqrt{3})}{-2}$

$\theta ={\tan }^{-1}(2-\sqrt{3})$

$\theta ={15}^o$