Question

A straight line makes $y$-intercept of 5. If the angle made by the line with $y$-axis is $60^\circ$ and the line intersects $x$-axis in the negative direction, then the equation of the line is

• A. $x + \sqrt{3}y + 5\sqrt{3} = 0$
• B. $x - \sqrt{3}y + 5\sqrt{3} = 0$
• C. $\sqrt{3}x - y + 5 = 0$
• D. $\sqrt{3}x + y + 5 = 0$
• E. $\sqrt{3}x - y + 5\sqrt{3} = 0$

Hint:

Angle with $y$-axis gives slope using $m = -\cot\theta$.

Answer 1

The Correct Option is B

Concept:
• Angle with $y$-axis = $60^\circ \Rightarrow$ slope $m = -\cot 60^\circ$
• Equation: $y = mx + c$

Step 1: Find slope
\[ m = -\cot 60^\circ = -\frac{1}{\sqrt{3}} \]

Step 2: Use intercept form
\[ y = -\frac{1}{\sqrt{3}}x + 5 \]

Step 3: Convert to standard form
\[ \sqrt{3}y = -x + 5\sqrt{3} \] \[ x - \sqrt{3}y + 5\sqrt{3} = 0 \]

Step 4: Check direction
Slope is negative $\Rightarrow$ intersects $x$-axis in negative direction \checkmark Final Conclusion:
Option (B)