Question

A straight line makes an angle a with the positive direction of x-axis, where \(\cos a=\frac{\sqrt3}{2}\). If it passes through (0,-2), then its equation is

• A. √3x+y+2=0
• B. √3y+x+2=0
• C. √3y+x+2√3=0
• D. √3y-x+2√3=0
• E. √3x-y-2√3=0

Answer 1

The Correct Option is D

Step 1: Find the slope of the line  
The given information states: \[ \cos a = \frac{\sqrt{3}}{2} \] Using the identity \( \sin^2 a + \cos^2 a = 1 \): \[ \sin^2 a = 1 - \left(\frac{\sqrt{3}}{2}\right)^2 = 1 - \frac{3}{4} = \frac{1}{4} \] \[ \sin a = \frac{1}{2} \] The slope \( m \) of the line is given by: \[ m = \tan a = \frac{\sin a}{\cos a} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

Step 2: Use the point-slope form 
The equation of a line passing through \( (x_1, y_1) \) with slope \( m \) is: \[ y - y_1 = m(x - x_1) \] Given point \( (0, -2) \) and \( m = \frac{1}{\sqrt{3}} \), we get: \[ y + 2 = \frac{1}{\sqrt{3}} x \] Multiplying both sides by \( \sqrt{3} \) to clear the fraction: \[ \sqrt{3} y + 2\sqrt{3} = x \] Rearranging: \[ \sqrt{3} y - x + 2\sqrt{3} = 0 \]

Final Answer: The equation of the line is \(\sqrt{3}y - x + 2\sqrt{3} = 0\).