in coordinate geometry while finding the angle bisectors of a pair of straight lines will the obtuse angle bisector always have - sign

In coordinate geometry, when finding the angle bisectors of a pair of straight lines, the obtuse angle bisector does indeed always have a negative sign in its equation.

### Explanation:

When you are given the equations of two straight lines, say \( L_1: a_1x + b_1y + c_1 = 0 \) and \( L_2: a_2x + b_2y + c_2 = 0 \), the equations of the angle bisectors are derived using the following formula:

\[
\frac{a_1x + b_1y + c_1}{\sqrt{a_1^2 + b_1^2}} \pm \frac{a_2x + b_2y + c_2}{\sqrt{a_2^2 + b_2^2}} = 0
\]

This equation gives you two angle bisectors:

1. **The bisector of the acute angle** (associated with the positive sign).
2. **The bisector of the obtuse angle** (associated with the negative sign).

### Why Does the Obtuse Angle Bisector Have a Negative Sign?

- The acute angle bisector is the one that lies between the two lines, making a smaller angle with both lines. Therefore, it is associated with the positive sign in the equation.
  
- The obtuse angle bisector, on the other hand, lies outside the acute angle, making a larger angle with one of the lines. This bisector is associated with the negative sign in the equation because it corresponds to the larger (obtuse) angle between the two lines.

### Summary:
In coordinate geometry, when determining the angle bisectors of two straight lines, the bisector of the obtuse angle will always have a negative sign in its equation.

Would you like more details or have any questions about this? Here are some related questions that might interest you:

1. How do you derive the equation for the angle bisectors between two lines?
2. What is the geometric significance of the positive and negative signs in angle bisectors?
3. Can you have multiple obtuse angle bisectors for different pairs of lines?
4. How does the angle between two lines affect the signs in their angle bisector equations?
5. How do the slopes of the lines influence the angle bisectors?

**Tip**: Always carefully consider the signs when working with angle bisectors in coordinate geometry, as they directly correspond to the acute and obtuse angles formed by the intersecting lines.

Explore how to find angle bisectors of two lines in coordinate geometry. Learn about the equations involved, including why the obtuse angle bisector always has a negative sign. Suitable for high school students studying coordinate geometry.

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