To determine the condition for a point [tex]\((a, b)\)[/tex] to lie on the x-axis, we need to consider the defining property of the x-axis: any point on the x-axis has a [tex]\(y\)[/tex]-coordinate of 0. This means that for the point [tex]\((a, b)\)[/tex] to be on the x-axis, [tex]\(b\)[/tex] must be 0.
Let's examine each of the given options to see which one satisfies this condition.
a) [tex]\(a^x=0, y=0\)[/tex]
This condition is not correctly formulated as it introduces an ambiguous term [tex]\(a^x\)[/tex], and incorrectly uses [tex]\(y\)[/tex] instead of [tex]\(b\)[/tex]. This option is incorrect.
c) [tex]\(a^2 \neq 0, \quad \stackrel{x}{b}=0\)[/tex]
This option can be read as stating two conditions:
- [tex]\(a^2 \neq 0\)[/tex]: which implies that [tex]\(a\)[/tex] is not zero.
- [tex]\(\stackrel{x}{b}=0\)[/tex]: suggesting that [tex]\(b=0\)[/tex].
Since [tex]\(b=0\)[/tex] is the requirement for the point to lie on the x-axis, this condition is correct. The [tex]\(a^2 \neq 0\)[/tex] condition simply restates that [tex]\(a\)[/tex] should not be zero, which is not strictly necessary but does not contradict the requirement for [tex]\(b\)[/tex] to be zero.
b) [tex]\(a \neq 0, b \neq 0\)[/tex]
This condition indicates that both [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are non-zero. However, [tex]\(b\)[/tex] must be 0 for the point to lie on the x-axis. Hence, this option is incorrect.
d) [tex]\(a=0, \quad b \neq 0\)[/tex]
This condition indicates that [tex]\(a\)[/tex] is zero and [tex]\(b\)[/tex] is non-zero. Such a point [tex]\((0, b)\)[/tex] does not lie on the x-axis because the [tex]\(y\)[/tex]-coordinate [tex]\(b\)[/tex] is not zero. This option is also incorrect.
From the analysis, the correct condition for the point [tex]\((a, b)\)[/tex] to lie on the x-axis is:
c) [tex]\(a^2 \neq 0, \quad \stackrel{x}{b}=0\)[/tex]
So, the correct answer is option 2 (c).
