Answer :
To solve this problem, we need to focus on the concept of the angle between two lines and how it relates to their slopes.
Understanding the X-intercept:
- The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. For line [tex]l_1[/tex], the x-intercept is given as [tex]1[/tex]. This means that line [tex]l_1[/tex] passes through the point [tex](1, 0)[/tex].
Slope of Line [tex]l_1[/tex]:
- In general terms, if a line has an x-intercept [tex]a[/tex] and passes through the origin, its equation can be expressed as [tex]y = m(x - a)[/tex], where [tex]m[/tex] is the slope.
- Given that the x-intercept is [tex]1[/tex], the line equation takes the form [tex]y = m(x - 1)[/tex].
- However, since the problem doesn't explicitly give more information about [tex]l_1[/tex], we will assume it has the standard form you have derived, [tex]y = m(x - 1)[/tex].
Angle Between Two Lines:
- The tangent of the angle [tex]\theta[/tex] between two lines with slopes [tex]m_1[/tex] and [tex]m_2[/tex] is given by the formula:
[tex]\tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1m_2} \right|[/tex] - The question specifies this angle as [tex]-1[/tex], which indicates an angle measure in radians or may imply certain specific slopes; however, it is a bit abstract without context.
- The tangent of the angle [tex]\theta[/tex] between two lines with slopes [tex]m_1[/tex] and [tex]m_2[/tex] is given by the formula:
Finding [tex]\tan(\theta)[/tex]:
- Since the question provides minimal information beyond requesting [tex]\tan(\theta)[/tex] itself, it's limited in presentation. Typically, additional information about the slope of [tex]l_2[/tex] or its relation to [tex]l_1[/tex] would be necessary to find [tex]\tan(\theta)[/tex].
Conclusion:
- Without explicit slopes for both lines or the context of what [tex]-1[/tex] indicates precisely in terms of geometric configuration, the presentation has inherent constraints for an accurate, quantified solution.
For better assistance or derivations, typically knowing additional angles or slope measures would suffice, which isn't presented at face value in the current question.
