Find the equation of the line which makes an angle of 30° with the positive direction of the x-axis and cuts off an intercept of 4 units with the negative direction of the y - axis ?

  1. \(x + \sqrt 3 y + 4\sqrt 3 = 0\)
  2. \(x + \sqrt 3 y - 4\sqrt 3 = 0\)
  3. \(x - \sqrt 3 y + 4\sqrt 3 = 0\)
  4. \(x - \sqrt 3 y - 4\sqrt 3 = 0\)

Answer (Detailed Solution Below)

Option 4 : \(x - \sqrt 3 y - 4\sqrt 3 = 0\)
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NDA 01/2025: English Subject Test
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30 Questions 120 Marks 30 Mins

Detailed Solution

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CONCEPT:

The equation of a line whose slope is m and which makes an intercept c on the Y-axis is given by: y = mx + c

CALCULATION:

Here, we have to find the equation of the line which makes an angle of 30° with the positive direction of the x-axis and cuts off an intercept of 4 units with the negative direction of the y - axis

As we know that, equation of a line whose slope is m and which makes an intercept c on the Y-axis is given by: y = mx + c

Here, we have c = - 4 and \(m = tan \ 30 = \frac{1}{\sqrt 3} \)

So, the equation of the required line is: \(y = \frac{1}{\sqrt 3} \times x - 4\)

⇒ \(\sqrt 3 \ y = x - 4\sqrt 3\)
 
So, the equation of the required line is \(x - \sqrt 3 y - 4\sqrt 3 = 0\)
 
Hence, option D is the correct answer.
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