Question

The slope of the line through the origin which makes an angle of $30^\circ$ with the positive direction of Y-axis measured anticlockwise is :

• A. $-\frac{2}{3}$
• B. $-\sqrt{3}$
• C. $\frac{3}{2}$
• D. $-\frac{1}{\sqrt{3}}$

Hint:

Draw a quick mental sketch of the coordinate axes! A line tilted $30^\circ$ into the second quadrant is clearly sloping downwards from left to right, meaning its slope must be negative. Since it is steeper than a $45^\circ$ line, its absolute value must be greater than 1, which instantly points you directly to $-\sqrt{3}$ over $-\frac{1}{\sqrt{3}}$!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to determine the mathematical slope of a straight line passing through the origin that is oriented at an angle of $30^\circ$ relative to the positive y-axis, measured in the counter-clockwise (anticlockwise) direction.

Step 2: Key Formula or Approach: The slope $m$ of a straight line is defined by the tangent of its inclination angle $\theta$, where $\theta$ is the angle that the line makes with the

positive direction of the x-axis measured anticlockwise: $$ m = \tan\theta $$

Step 3: Detailed Explanation:
Let's trace out the angles geometrically starting from the standard coordinate layout:

• The positive direction of the y-axis is already located at an angle of $90^\circ$ relative to the positive direction of the x-axis.

• The problem states that our line is rotated an additional $30^\circ$ anticlockwise away from this positive y-axis.
Therefore, the total inclination angle $\theta$ of the line relative to the positive x-axis is: $$ \theta = 90^\circ + 30^\circ = 120^\circ $$ Now, we calculate the slope by taking the tangent of this total angle: $$ m = \tan(120^\circ) $$ Using the trigonometric reduction identity $\tan(180^\circ - \phi) = -\tan\phi$: $$ m = \tan(180^\circ - 60^\circ) = -\tan(60^\circ) $$ Substituting the exact standard value $\tan(60^\circ) = \sqrt{3}$: $$ m = -\sqrt{3} $$

Step 4: Final Answer:
The slope of the line is $-\sqrt{3}$, which corresponds to option (B).