Question

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

Answer 1

Answer: (A)

Let slope of the line be $m$.
Now, the equation of the line passing through $(1,0)$ with slope $m$ is
$y−0=m(x−1)$
$⇒mx−y−m=0$
Distance from origin to the given line $=\frac{\sqrt{3}}{2}$
$⇒\frac{âˆ\pounds −mâˆ\pounds }{\sqrt{1+m^2}}=\frac{\sqrt{3}}{2}$
$⇒4m^2=3\left(1+m^2\right)$
$⇒m^2=3$
$⇒m=Â\pm \sqrt{3}$
$∴$ Required equations are
$\sqrt{3}x−y−\sqrt{3}=0$ and
$−\sqrt{3}x−y+\sqrt{3}=0$
$⇒\sqrt{3}x−y−\sqrt{3}=0$ and
$\sqrt{3}x+y−\sqrt{3}=0$