Question

The angle between two lines is \(45^\circ\) and slope of one line is \( \frac{1}{4} \) then which is the possible value of the slope of the other line.

• A. \( \frac{5}{4} \)
• B. \( \frac{5}{3} \)
• C. \( \frac{4}{5} \)
• D. \( \frac{3}{5} \)

Hint:

Always use \( \tan \theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right| \) for angle between lines.

Answer 1

The Correct Option is B

Step 1: Use formula for angle between two lines.
\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]

Step 2: Substitute given values.
\[ \theta = 45^\circ,\quad m_1 = \frac{1}{4} \]
\[ \tan 45^\circ = 1 \]

Step 3: Apply formula.
\[ 1 = \left| \frac{m_2 - \frac{1}{4}}{1 + \frac{1}{4}m_2} \right| \]

Step 4: Remove modulus (consider positive case).
\[ \frac{m_2 - \frac{1}{4}}{1 + \frac{1}{4}m_2} = 1 \]

Step 5: Solve equation.
\[ m_2 - \frac{1}{4} = 1 + \frac{1}{4}m_2 \]
\[ m_2 - \frac{1}{4}m_2 = 1 + \frac{1}{4} \]
\[ \frac{3}{4}m_2 = \frac{5}{4} \]
\[ m_2 = \frac{5}{3} \]

Step 6: Check options.
\[ \frac{5}{3} \text{ is present} \]

Step 7: Final Answer.
\[ \boxed{\frac{5}{3}} \]