Problem Analysis

We are given three points: X(a, p), Y(b, q), and Z(c, r).

The x-coordinates \(a\), \(b\), and \(c\) are in Arithmetic Progression (AP). This means the difference between consecutive terms is constant:

\(b - a = c - b\)

Which simplifies to:

\(2b = a + c\)

We are also given two conditions:

1. The y-coordinates \(p\), \(q\), and \(r\) are not in AP.
2. The x-coordinate of Y is equal to the x-coordinate of Z: \(b = c\).

We need to find what the line joining X, Y, and Z is parallel to.

Coordinate Analysis

Let's use the given conditions:

• From the AP condition: \(2b = a + c\).
• From the specific condition: \(b = c\).

Substitute \(c = b\) into the AP equation:

\(2b = a + b\)

Subtract \(b\) from both sides:

\(b = a\)

So, we have found that \(a = b\). Since we were given \(b = c\), it follows that:

\(a = b = c\)

Line Geometry

This result means that all three points X, Y, and Z have the same x-coordinate.

• Point X is (a, p).
• Point Y is (b, q) = (a, q).
• Point Z is (c, r) = (a, r).

A line connecting points that share the same x-coordinate is a vertical line. The equation of this line is of the form \(x = k\), where \(k\) is the common x-coordinate (in this case, \(a\)).

Vertical lines are parallel to the y-axis.

The condition that \(p, q, r\) are not in AP ensures that the points are not necessarily equally spaced vertically, but it does not change the fact that the line containing them is vertical.

Conclusion

The line joining the points X, Y, and Z, where \(a = b = c\), is parallel to the y-axis.