Question

The angle between the lines y = 3x + 2 and y = 2x + 1

• A. $\tan^{-1}(\frac{1}{5})$
• B. $\tan^{-1}(\frac{1}{8})$
• C. $\tan^{-1}(\frac{1}{5})$
• D. $\tan^{-1}(1)$

Answer 1

Answer: (D)

$\tan^{-1}(1)$

Answer 2

To find the angle between two lines, we use the formula for the angle $\theta$ between two lines with slopes $m_1$ and $m_2$:

$\tan\theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$

The given lines are:

1. $y = 3x + 2$
2. $y = 2x + 1$

From the slope-intercept form $y = mx + c$, we can identify the slopes: For the first line, $m_1 = 3$. For the second line, $m_2 = 2$.

Now, substitute these values into the formula: $\tan\theta = \left| \frac{2 - 3}{1 + (3)(2)} \right|$ $\tan\theta = \left| \frac{-1}{1 + 6} \right|$ $\tan\theta = \left| \frac{-1}{7} \right|$ $\tan\theta = \frac{1}{7}$ So, the angle between the lines is $\theta = \tan^{-1}\left(\frac{1}{7}\right)$.

However, this calculated value $\tan^{-1}\left(\frac{1}{7}\right)$ is not among the options. This suggests a possible typo in the question or the options provided.

Let's consider a common typo that would lead to one of the options. If the second line was intended to be $y = -2x + 1$ instead of $y = 2x + 1$, then $m_2 = -2$.

Let's recalculate with $m_1 = 3$ and $m_2 = -2$: $\tan\theta = \left| \frac{-2 - 3}{1 + (3)(-2)} \right|$ $\tan\theta = \left| \frac{-5}{1 - 6} \right|$ $\tan\theta = \left| \frac{-5}{-5} \right|$ $\tan\theta = |1|$ $\tan\theta = 1$ In this case, the angle $\theta = \tan^{-1}(1)$. This matches Option 4. This is a common angle ($45^\circ$) and a common result in such problems, making it a highly probable intended answer given the options.

Therefore, assuming a likely typo in the question where the second line should have been $y = -2x + 1$, the angle between the lines is $\tan^{-1}(1)$.