Question

The equations of the lines passing through the point (1, 0) and at a distance `sqrt(3)/2` from the origin, are ______.

Options

• `sqrt(3)x + y - sqrt(3)` = 0, `sqrt(3)x - y - sqrt(3)` = 0

• `sqrt(3)x + y + sqrt(3)` = 0, `sqrt(3)x - y + sqrt(3)` = 0

• `x + sqrt(3)y - sqrt(3)` = 0, `x - sqrt(3)y - sqrt(3)` = 0

• None of these

Answer 1

The equations of the lines passing through the point (1, 0) and at a distance `sqrt(3)/2` from the origin, are `sqrt(3)x + y - sqrt(3)` = 0, `sqrt(3)x - y - sqrt(3)` = 0.

Explanation:

Equation of any line passing through (1, 0) is y – 0 = m(x – 1)

⇒ mx – y – m = 0

Distance of the line from origin is `sqrt(3)/2`

∴ `sqrt(3)/2 = |(m xx 0 - 0 - m)/sqrt(1 + m^2)|`

⇒ `sqrt(3)/2 = |(-m)/sqrt(1 + m^2)|`

Squaring both sides, we get

`3/4 = m^2/(1 + m^2)`

⇒ 4m² = 3 + 3m²

⇒ 4m² – 3m² = 3

⇒ m² = 3

∴ m = `+-  sqrt(3)`

∴ Required equations are `+- sqrt(3)x - y -+ sqrt(3)` = 0

i.e., `sqrt(3)x - y - sqrt(3)` = 0 and `- sqrt(3)x - y + sqrt(3)` = 0

⇒ `sqrt(3)x + y - sqrt(3)` = 0