A question says that a pair of st. Lines inclined at an angle theta. Does it mean that the angle between the two lines is theta or they make an angle theta with the x-axis?

When you come across the phrase "a pair of straight lines inclined at an angle theta," it typically refers to the angle between the two lines themselves, not the angle each line makes with the x-axis. This distinction is crucial for understanding the geometric relationships involved.

Understanding the Concept of Inclination

To clarify, let's break down what it means for lines to be inclined at an angle. When we say two lines are inclined at an angle theta, we are specifically talking about the angle formed where the two lines intersect. This angle is measured between the two lines, not in relation to any axes.

Visualizing the Lines

Imagine you have two lines on a coordinate plane. If you draw them so that they intersect, the angle formed at the intersection point is what we refer to as theta. For example, if one line is slanted upwards to the right and the other line is slanted downwards to the right, the angle between them at their intersection is theta.

Example for Clarity

Consider two lines:

• Line 1: y = m1 * x + c1
• Line 2: y = m2 * x + c2

Here, m1 and m2 are the slopes of the lines. The angle theta between these two lines can be calculated using the formula:

tan(theta) = |(m2 - m1) / (1 + m1 * m2)|

This formula shows that the angle theta is derived from the slopes of the lines, further emphasizing that it is the angle between the lines themselves.

Distinction with Axes

On the other hand, if we were discussing the angles that each line makes with the x-axis, we would refer to those angles separately. For instance, if Line 1 makes an angle alpha with the x-axis and Line 2 makes an angle beta, then the relationship between these angles and the angle between the lines can be expressed as:

theta = |alpha - beta|

This highlights that while the inclination of the lines is theta, the angles with the x-axis are distinct and can be calculated separately.

Conclusion

In summary, when you see "a pair of straight lines inclined at an angle theta," it specifically means the angle between the two lines themselves. Understanding this distinction is essential for solving problems related to geometry and line equations effectively.