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. \( \sqrt{3}x + y - \sqrt{3} = 0, \; \sqrt{3}x - y - \sqrt{3} = 0 \)
• 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 the above

Hint:

Use slope form for line through a point, then apply distance formula to find possible slopes.

Answer 1

The Correct Option is A

Concept: Distance of a line \( Ax + By + C = 0 \) from origin: \[ \frac{|C|}{\sqrt{A^2 + B^2}} \]

Step 1: Equation of line through \( (1,0) \). \[ y = m(x-1) \Rightarrow mx - y - m = 0 \]

Step 2: Distance from origin: \[ \frac{| -m |}{\sqrt{m^2 + 1}} = \frac{\sqrt{3}}{2} \] \[ \frac{|m|}{\sqrt{m^2 + 1}} = \frac{\sqrt{3}}{2} \]

Step 3: Solve: \[ \frac{m^2}{m^2 + 1} = \frac{3}{4} \Rightarrow 4m^2 = 3m^2 + 3 \Rightarrow m^2 = 3 \] \[ m = \pm \sqrt{3} \]

Step 4: Substitute values: \[ y = \sqrt{3}(x-1), \quad y = -\sqrt{3}(x-1) \] \[ \sqrt{3}x - y - \sqrt{3} = 0, \quad \sqrt{3}x + y - \sqrt{3} = 0 \] Final Answer: \[ \text{Option (A)} \]