Question

The equation of the straight line through the origin making angle $\varphi$ with the line $y=mx+b$, is

• A. $y=\left(\frac{m+\tan \varphi }{1-m\tan \varphi }\right)x$
• B. $y=\left(\frac{m-\tan \varphi }{1+\tan \varphi }\right)x$
• C. $y=\left(\frac{\tan \varphi }{1+m\tan \varphi }\right)x$
• D. $y=\left(\frac{m-\tan \varphi }{1+m\tan \varphi }\right)x$

Answer 1

Answer: (A), (D)

The correct options are
A $y=\left(\frac{m+\tan \varphi }{1-m\tan \varphi }\right)x$
D $y=\left(\frac{m-\tan \varphi }{1+m\tan \varphi }\right)x$

Since, required line makes an angle of $\varphi$ with $y=mx+b$
Let $m=\tan \theta$
Hence, the required line makes $\theta \pm \varphi$ angle with $x-$ axis.

Now,
$\tan (\theta \pm \varphi )=\frac{m\pm \tan \varphi }{1\mp m\tan \varphi }$

So, equation of required line will be
$y=\left(\frac{m\pm \tan \varphi }{1\mp m\tan \varphi }\right)x$