Question

The equations of the lines passing through the point $(1,0)$ and at a distance $\frac{\sqrt{3}}{2}$ from the origin are

• A. $\sqrt{3}x+y-\sqrt{3}=0,\sqrt{3}x-y-\sqrt{3}=0$
• B. $\sqrt{3}x+y+\sqrt{3}=0,\sqrt{3}x-y+\sqrt{3}=0$
• C. $x+\sqrt{3}y-\sqrt{3}=0,x-\sqrt{3}y-\sqrt{3}=0$
• D. None of the above

Answer 1

The correct option is A $\sqrt{3}x+y-\sqrt{3}=0,\sqrt{3}x-y-\sqrt{3}=0$

Let slope of a line be $m$.
Now, the equation of a line passing through $(1,0)$ is
$y-0=m(x-1)$
$\Rightarrow mx-y-m=0$
Distance from origin $=\frac{\sqrt{3}}{2}$
$\Rightarrow \frac{|-m|}{\sqrt{1+m^2}}=\frac{\sqrt{3}}{2}$
$\Rightarrow 4m^2=3(1+m^2)$
$\Rightarrow m^2=3$
$\Rightarrow m=\pm \sqrt{3}$
Therefore, equations of lines are
$\sqrt{3}x-y-\sqrt{3}=0$ and $-\sqrt{3}x-y+\sqrt{3}=0$
$\Rightarrow \sqrt{3}x-y-\sqrt{3}=0$ and $\sqrt{3}x+y-\sqrt{3}=0$.