Question

Find the equation of the straight line passing through the origin and making an angle of ${45}^o$ with the straight line $\sqrt{3}x+y=11$.

• A. $y=(\sqrt{2}\pm 2)x$
• B. $y=(\sqrt{3}\pm 2)x$
• C. $y=(\sqrt{2}\pm 3)x$
• D. $y=(\sqrt{3}\pm 3)x$

Answer 1

The correct option is B $y=(\sqrt{3}\pm 2)x$

As we know, the equation of two lines passing through a point $(x,y)$ and making an angle $\theta$ with the given line $y=mx+C$ are

$y-y_1=\frac{m\pm \tan \theta }{1\pm m\tan \theta }(x-x_1)$

Here, $x_1=0,y=0$ ($\because$ lines passing hrough origin)

$\theta ={45}^o$ and $m=-\sqrt{3}$

So, the equations of the required lines are

$y-0=\frac{-\sqrt{3}+\tan {45}^o}{1+\sqrt{3}\tan {45}^o}(x-0)$ and $y-0=\frac{-\sqrt{3}-\tan {45}^o}{1-\sqrt{3}\tan {45}^o}(x-0)$

$\Rightarrow y=\frac{-\sqrt{3}+1}{1+\sqrt{3}}x$ and $y=\frac{\sqrt{3}+1}{\sqrt{3}-1}x$

$\Rightarrow y=\frac{(-\sqrt{3}+1)(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3}}x$ and

$y=\frac{(\sqrt{3}+1)(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}x$

$\Rightarrow y=(\sqrt{3}-2)x$ and $y=(\sqrt{3}+2)x$

Hence, the equation of the straight line passing through the origin and making an angle of ${45}^o$ with $\sqrt{3}x+y=11$

is $y=(\sqrt{3}\pm 2)x$