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Q.
The equations of the lines passing through the point $(1,0)$ and at a distance of $\frac{\sqrt{3}}{2}$ from the origin, are
Straight Lines
Solution:
Let slope of the line be $m$.
Now, the equation of the line passing through $(1,0)$ with slope $m$ is
$y - 0 = m(x- 1)$
$\Rightarrow mx -y - m = 0$
Distance from origin to the given line $=\frac{\sqrt{3}}{2}$
$\Rightarrow \frac{\left|-m\right|}{\sqrt{1+m^{2}}}=\frac{\sqrt{3}}{2}$
$\Rightarrow 4m^{2}=3\left(1+m^{2}\right)$
$\Rightarrow m^{2}=3$
$\Rightarrow m=\pm\sqrt{3}$
$\therefore $ Required equations 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
$\sqrt{3}x+y-\sqrt{3}=0$