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) $\sqrt{3}x\pm y–\sqrt{3}=0$

(b) $\sqrt{3}x\pm y+\sqrt{3}=0$

(c) $x\pm \sqrt{3}y–\sqrt{3}=0$

(d) none of these

Answer 1

Let us suppose slope of the line is m and it is given that line passes through (1, 0)
∴ equation of line is y – 0 = m(x – 1)
i.e y = mx – m
Also given distance between (0, 0) and y = mx – m is $\frac{\sqrt{3}}{2}$
$$\begin{gathered} \mathrm{i}.\mathrm{e}\frac{\sqrt{3}}{2}=\frac{\left|0-0+m\right|}{\sqrt{1+m^2}} \\ \mathrm{i}.\mathrm{e}\frac{\sqrt{3}}{2}=\frac{\left|m\right|}{\sqrt{1+m^2}} \\ \mathrm{i}.\mathrm{e}\frac{3}{2}=\frac{m^2}{1+m^2} \\ \mathrm{i}.\mathrm{e}3+3m^2=2m^2 \\ \mathrm{i}.\mathrm{e}{m}^2=3 \\ \mathrm{i}.\mathrm{e}m=\pm \sqrt{3} \\ \therefore \mathrm{equation}\mathrm{of}\mathrm{line}\mathrm{is}\sqrt{3}x+y-\sqrt{3}=0\mathrm{or}\sqrt{3}x-y-\sqrt{3}=0 \end{gathered}$$

Hence, the correct answer is option A.