The problem asks to find the angle between two lines given their equations:

1. $l_1: 5x - y - 6 = 0$
2. $l_2: y + \frac{1}{5}x - \frac{17}{5} = 0$

Solution:

Step 1: Rewrite the equations in slope-intercept form ($y = mx + b$):

1. For $l_1: 5x - y - 6 = 0$:
   Rearrange to solve for $y$:
   $y = 5x - 6$
   Here, the slope ($m_1$) is $5$.

2. For $l_2: y + \frac{1}{5}x - \frac{17}{5} = 0$:
   Rearrange to solve for $y$:
   $y = -\frac{1}{5}x + \frac{17}{5}$
   Here, the slope ($m_2$) is $-\frac{1}{5}$.

Step 2: Use the formula for the angle between two lines:

The angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by:
$\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right|$

Substitute $m_1 = 5$ and $m_2 = -\frac{1}{5}$:
$\tan\theta = \left| \frac{5 - \left(-\frac{1}{5}\right)}{1 + 5\left(-\frac{1}{5}\right)} \right|$

Simplify: $\tan\theta = \left| \frac{5 + \frac{1}{5}}{1 - 1} \right|$

The denominator becomes $0$, indicating that the lines are perpendicular. Thus, the angle $\theta = 90^\circ$.

Final Answer:

The angle between the lines is $90^\circ$.

Would you like a detailed derivation or have any questions?

Related Questions:

1. How do you find the slopes of lines from their equations?
2. What is the geometric meaning of perpendicular lines?
3. Can you calculate the angle between lines if their slopes are undefined?
4. What does it mean if the denominator in $\tan\theta$ is zero?
5. How does the formula for $\tan\theta$ change if one line is vertical?

Tip:

Always verify the slopes carefully, as small errors can lead to incorrect conclusions about angles.