To determine the angle that the line [tex]\( x \cos \alpha + y \sin \alpha = p \)[/tex] makes with the positive [tex]\( x \)[/tex]-axis, we need to carefully consider the standard form of a linear equation and the associated trigonometric relationships.
First, recall that the general form of a line equation in Cartesian coordinates is:
[tex]\[ Ax + By + C = 0 \][/tex]
In our given line equation, we can rewrite it to the standard form:
[tex]\[ x \cos \alpha + y \sin \alpha = p \\
or,
x \cos \alpha + y \sin \alpha - p = 0 \][/tex]
Here, [tex]\( A = \cos \alpha \)[/tex] and [tex]\( B = \sin \alpha \)[/tex].
The angle [tex]\( \theta \)[/tex] that a line makes with the positive [tex]\( x \)[/tex]-axis can be found using the relationship:
[tex]\[ \tan \theta = -\frac{A}{B} \][/tex]
For the given line equation:
[tex]\[ \tan \theta = -\frac{\cos \alpha}{\sin \alpha} = -\cot \alpha \][/tex]
The line's slope [tex]\( m \)[/tex] is therefore:
[tex]\[ m = -\cot \alpha \][/tex]
Thus, the angle [tex]\( \theta \)[/tex] can be determined as:
[tex]\[ \theta = \alpha \][/tex]
Therefore, the correct option which indicates the angle this line makes with the positive [tex]\( x \)[/tex]-axis is:
[tex]\[ \boxed{\alpha} \][/tex]
Hence, the correct answer is (6).
