Question

The acute angle between the line $4x-2y+13=0$ and the line which makes equal intercepts with the co-ordinate axes is

• A. $\tan^{-1}(2)$
• B. $\tan^{-1}(3)$
• C. $\tan^{-1}(\frac{1}{2})$
• D. $\tan^{-1}(\frac{1}{3})$

Hint:

Lines with equal intercepts have a slope of $-1$.

Answer 1

The Correct Option is B

Step 1: Concept
Angle between two lines with slopes $m_1$ and $m_2$ is $\tan \theta = |\frac{m_1 - m_2}{1 + m_1 m_2}|$.

Step 2: Analysis
- Line 1: $4x-2y+13=0 \implies y = 2x + \frac{13}{2}$. So, $m_1 = 2$. - Line 2: Equal intercepts means $x/a + y/a = 1 \implies x+y=a$. So, $m_2 = -1$.

Step 3: Calculation
- $\tan \theta = |\frac{2 - (-1)}{1 + (2)(-1)}| = |\frac{3}{1 - 2}| = |-3| = 3$.

Step 4: Conclusion
$\theta = \tan^{-1}(3)$. Final Answer: (B)