To solve this problem, let's analyze the given conditions and break them down step-by-step.
1. Equal Distance from the [tex]\(y\)[/tex]-Axis:
When two points are equidistant from the [tex]\(y\)[/tex]-axis, it means their [tex]\(x\)[/tex]-coordinates have the same absolute value but opposite signs. For instance, if one point has an [tex]\(x\)[/tex]-coordinate of [tex]\(a\)[/tex], the other point will have an [tex]\(x\)[/tex]-coordinate of [tex]\(-a\)[/tex].
2. Parallel to the [tex]\(x\)[/tex]-Axis:
If the line joining the two points is parallel to the [tex]\(x\)[/tex]-axis, it means they have the same [tex]\(y\)[/tex]-coordinate. Let's denote the shared [tex]\(y\)[/tex]-coordinate as [tex]\(y\)[/tex].
Let's denote the coordinates of the points as [tex]\((a, y)\)[/tex] and [tex]\((-a, y)\)[/tex].
3. Sum of Their Abscissas:
The abscissa of a point is its [tex]\(x\)[/tex]-coordinate. We need to find the sum of the [tex]\(x\)[/tex]-coordinates (abscissas) of these points.
For the points [tex]\((a, y)\)[/tex] and [tex]\((-a, y)\)[/tex], their abscissas are [tex]\(a\)[/tex] and [tex]\(-a\)[/tex] respectively. Therefore, the sum of their abscissas is:
[tex]\[
a + (-a)
\][/tex]
4. Calculate the Sum:
[tex]\[
a + (-a) = 0
\][/tex]
Therefore, the sum of the abscissas of the two points is [tex]\(0\)[/tex].
