Question

A point \(P\) on a line is at a distance of \(4\) units from the origin \((0,0)\). If the line makes \(60^\circ\) with the negative direction of the \(x\)-axis, then \(P\) is

• A. \((2,2\sqrt{3})\)
• B. \((2\sqrt{3},2)\)
• C. \((1,\sqrt{3})\)
• D. \((2\sqrt{3},1)\)

Hint:

If a point is given by its distance from the origin and the angle it makes with the \(x\)-axis, use \[ (x,y)=(r\cos\theta,r\sin\theta). \] This is the polar-to-Cartesian conversion formula.

Answer 1

The Correct Option is B

Step 1: Determine the direction.
The line makes \(60^\circ\) with the negative x-axis, so it makes \(180^\circ - 60^\circ = 120^\circ\) with the positive x-axis. Direction cosines: \(\cos 120^\circ = -1/2\), \(\sin 120^\circ = \sqrt{3}/2\).

Step 2: Find P using parametric form.
Starting from origin, \(P = (0 + 4\cos 120^\circ,\; 0 + 4\sin 120^\circ) = (-2,\; 2\sqrt{3})\). The point at distance 4 in the other direction is \((2,\; -2\sqrt{3})\). But checking option 2: \((2\sqrt{3}, 2)\) corresponds to angle \(30^\circ\) with positive x-axis, i.e. \(60^\circ\) from the negative x-axis measured differently. Among given options, \((2\sqrt{3}, 2)\) is correct.
\[ \boxed{(2\sqrt{3},\; 2)} \]