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# Choose the correct answer from the given four options. The equations of the lines passing through the point (1, 0) and at a distance $\large\frac{\sqrt 3 }{2}$ from the origin, are

$\begin {array} {1 1} (A)\;\sqrt 3 x + y - \sqrt 3 = 0, \sqrt 3 x - y - \sqrt 3 = 0 & \quad (B)\;\sqrt 3 x + y + \sqrt 3 = 0, \sqrt 3 x - y + \sqrt 3 = 0 \\ (C)\;x+ \sqrt 3 \: y - \sqrt 3 =0, x - \sqrt 3 y - \sqrt 3 = 0 & \quad (D)\;\text{None of these} \end {array}$ Comment
A)
Toolbox:
• Distance of a line $ax+by+c=0$ from the origin (0,0) is $d = \bigg| \large\frac{c}{\sqrt{A^2+B^2}} \bigg|$
Step 1 :
It is given tht the line passes through the point (1,0) and its distance from the origin is $\large\frac{\sqrt 3 }{2}$
Consider the option 'A'
$\sqrt 3 (1)+(0)-\sqrt 3 =0, \: \sqrt 3 (1) - 0 - \sqrt 3 =0$
LHS = RHS
Also the distance of this line from origin is $d = \bigg| \large\frac{-\sqrt 3 }{\sqrt{(\sqrt3)^2+(1)^2}} \bigg|$$= \large\frac{\sqrt 3 }{2}$
Hence option 'A' satisfies the condition .
$\therefore$ Option 'A' is the correct answer.