Question

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

• A. $\sqrt{3}x+y-\sqrt{3}=0$
• B. $x+\sqrt{3}y-\sqrt{3}=0$
• C. $\sqrt{3}x-y-\sqrt{3}=0$
• D. $x-\sqrt{3}y-\sqrt{3}=0$

Answer 1

Answer: (A), (C)

The correct options are
A $\sqrt{3}x+y-\sqrt{3}=0$
C $\sqrt{3}x-y-\sqrt{3}=0$

Equation of a line through $(1,0)$ is$y=m(x-1)$

$⟹y-mx+m=0$ using the formula of perpendicular distance of any point $(x_0,y_0)$ from line $ax+by+c=0$ is $\frac{|a{x}_0+b{y}_0+c|}{\sqrt{a^2+b^2}}$ perpendicular distance from origin (0,0) is

$|\frac{m}{\sqrt{m^2+1}}|=\frac{\sqrt{3}}{2}$

$⟹4m^2=3(1+m^2)⟹m^2=3⟹m=\pm \sqrt{3}$
Therefore, equation of lines are $\sqrt{3}x+y-\sqrt{3}=0$ and $\sqrt{3}x-y-\sqrt{3}=0$

Hence, the correct options are 'A' and 'C'.