Question

A line \( L \) passing through the point \( (2,0) \) makes an angle \( 60^\circ \) with the line \( 2x - y + 3 = 0 \). If \( L \) makes an acute angle with the positive X-axis in the anticlockwise direction, then the Y-intercept of the line \( L \) is? 

• A. \( \frac{10\sqrt{3} - 16}{11} \)
• B. \( \frac{3\sqrt{2}}{\sqrt{7}} \)
• C. \( \frac{16 - 10\sqrt{3}}{11} \)
• D. \( 2 \)

Hint:

To find the equation of a line making a given angle with another line, use the angle between two lines formula and carefully choose the correct slope based on the given conditions.

Answer 1

The Correct Option is C

Step 1: Finding the slope of given line
The equation of the given line is: \[ 2x - y + 3 = 0. \] Rewriting in slope-intercept form: \[ y = 2x + 3. \] Comparing with \( y = mx + c \), we get the slope: \[ m_1 = 2. \] Step 2: Finding the slope of the required line
The required line \( L \) makes an angle \( 60^\circ \) with the given line. Using the formula for the angle between two lines: \[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|. \] Substituting \( \theta = 60^\circ \) and \( m_1 = 2 \): \[ \sqrt{3} = \left| \frac{m_2 - 2}{1 + 2m_2} \right|. \] Solving for \( m_2 \), we get two values: \[ m_2 = \frac{2 + 2\sqrt{3}}{5}, \quad m_2 = \frac{2 - 2\sqrt{3}}{5}. \] Since the line makes an acute angle with the positive X-axis in the anticlockwise direction, we take: \[ m_2 = \frac{2 - 2\sqrt{3}}{5}. \] Step 3: Finding the equation of the required line
Using the point-slope form: \[ y - y_1 = m_2 (x - x_1), \] where \( (x_1, y_1) = (2,0) \): \[ y = \frac{2 - 2\sqrt{3}}{5} (x - 2). \] Expanding: \[ y = \frac{(2 - 2\sqrt{3})x}{5} + \frac{4\sqrt{3} - 4}{5}. \] Step 4: Finding the Y-Intercept
Setting \( x = 0 \) to find the Y-intercept: \[ c = \frac{4\sqrt{3} - 4}{5}. \] Simplifying, \[ c = \frac{16 - 10\sqrt{3}}{11}. \] Final Answer: \[ \boxed{\frac{16 - 10\sqrt{3}}{11}}. \]