Question

The axes being inclined at an angle of 60${}^o$, the angle between the two straight lines $y=2x+5$ and $2y+x+7=0$ is

• A. ${90}^{\circ }$
• B. ${\tan }^{-1}\frac{5}{3}$
• C. ${\tan }^{-1}\frac{\sqrt{3}}{2}$
• D. ${\tan }^{-1}\frac{5}{\sqrt{3}}$

Answer 1

The correct option is D ${\tan }^{-1}\frac{5}{\sqrt{3}}$

Consider equation (i)

$y=2x+5$

Therefore, $\tan \theta =\frac{m\sin \omega }{1+m\cos \omega }$

$=\frac{2\left(\frac{\sqrt{3}}{2}\right)}{1+\left(2\right)\frac{1}{2}}$

$=\frac{\sqrt{3}}{2}$ ...(a)

Consider equation (i)

$2y=-x-7$

Therefore, $\tan \theta =\frac{m\sin \omega }{1+m\cos \omega }$

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

$=\frac{-1}{\sqrt{3}}$ ...(b)

Now,

$M=\left|\frac{a-b}{1+ab}\right|$

$=\left|\frac{\frac{5}{2\sqrt{3}}}{\frac{1}{2}}\right|$

$=\left|\frac{5}{\sqrt{3}}\right|$