Consider the following statements in respect of the line passing through origin and inclining at an angle of 75° with the positive direction of x-axis :

1. The line passes through the point \(\left(1, \frac{1}{2−\sqrt{3}}\right)\).

2. The line entirely lies in first and third quadrants.

Which of the statements given above is/are correct ?

Concept:

The equation of the line passing through (h,k) and making an angle θ with the positive direction of the x-axis is (y - k) = tan θ (x - a) where tan θ is the slope of the line.

Value used:

• tan 75° = \(\sqrt 3 + 1 \over \sqrt 3 - 1\)

Calculation:

Given, the line passing through origin and inclining at an angle of 75° with the positive direction of x-axis.

1. The line passes through the point \(\left(1, \frac{1}{2−\sqrt{3}}\right)\).

The equation of the line passing through the origin (0,0) and inclining at an angle of 75° with the positive direction of the x-axis is (y - 0) = tan 75°(x - 0)

⇒ y = \(\sqrt 3 + 1 \over \sqrt 3 - 1\) x

⇒ \((\sqrt3 +1)x - (\sqrt 3 - 1) y =0\)

If line passes through a point then the point satisfies the equation,

For \(\left(1, \frac{1}{2−\sqrt{3}}\right)\),

 \((\sqrt3 +1)(1) - (\sqrt 3 - 1) ({1 \over 2 - \sqrt 3}) = {(\sqrt3 +1) (2 - \sqrt 3) - \sqrt 3 + 1\over 2 - \sqrt 3}\)

⇒ \({2\sqrt3 +2 - 3 -\sqrt 3- \sqrt 3 + 1\over 2 - \sqrt 3}\) = 0

⇒ \(\left(1, \frac{1}{2−\sqrt{3}}\right)\)lies on the line \((\sqrt3 +1)x - (\sqrt 3 - 1) y =0\)

So statement 1 is true.

2. The line entirely lies in the first and third quadrants.

The line passes through the origin (0,0) and inclining at an angle of 75° with the positive direction of the x-axis

Plotting the line on the graph from this information

It can clearly be seen that line only passes through first and third quadrant.

So, statement 2 is true.

∴ The correct option is (3).