Problem Analysis:

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

We are also given two conditions:

• The x-coordinates (a, b, c) are in Arithmetic Progression (AP).
• The y-coordinates (p, q, r) are in Arithmetic Progression (AP).

We need to determine the geometric arrangement of these points based on these conditions.

Condition Analysis: Arithmetic Progression

The condition that \(a, b, c\) are in AP implies:

\( b - a = c - b \quad \Rightarrow \quad 2b = a + c \)

Similarly, the condition that \(p, q, r\) are in AP implies:

\( q - p = r - q \quad \Rightarrow \quad 2q = p + r \)

Geometric Interpretation: Collinearity Check

To check if the points lie on a straight line (are collinear), we compare the slopes between consecutive pairs of points (XY and YZ).

The slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Slope between X(a, p) and Y(b, q):

\( m_{XY} = \frac{q-p}{b-a} \)

Slope between Y(b, q) and Z(c, r):

\( m_{YZ} = \frac{r-q}{c-b} \)

Relating Slopes using AP Conditions

From the AP conditions:

• \(2b = a + c \implies b - a = c - b\). Let this common difference be \(k\). Then \(b-a = k\) and \(c-b = k\).
• \(2q = p + r \implies q - p = r - q\). Let this common difference be \(m\). Then \(q-p = m\) and \(r-q = m\).

Now, substitute these back into the slope formulas:

\( m_{XY} = \frac{m}{k} \)

\( m_{YZ} = \frac{m}{k} \)

Since \(m_{XY} = m_{YZ}\), the slopes are equal.

Conclusion

When the slopes between consecutive points are equal, it means the points lie on the same straight line.

Therefore, the points X, Y, and Z are on a straight line.