If \(\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}\) and \(\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3}\) are in geometric progression and their common ratios are equal, then the points \(\mathrm{A}\left(\mathrm{a}_{1}, \mathrm{~b}_{1}\right)\) \(\mathrm{B}\left(\mathrm{a}_{2}, \mathrm{~b}_{2}\right)\) and \(\mathrm{C}\left(\mathrm{a}_{3}, \mathrm{~b}_{3}\right) \ldots \ldots\) (a) lie on the same line (b) lie on a circle (c) lie on an ellipse (d) None of these

A geometric progression (or geometric sequence) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This common ratio is an essential feature of geometric progressions and is central to understanding their behavior and properties.

For example, if we have three terms in a geometric progression: 2, 6, and 18, we say that the common ratio is 3, because each term is multiplied by 3 to get the next term (2 * 3 = 6, 6 * 3 = 18). Recognizing the common ratio allows us to generate any number of terms in the progression, calculate the sum of a finite number of terms, or even work backwards to find previous terms.

Understanding the concept of common ratios is crucial when dealing with problems that involve geometric progressions, as it provides a simple way to describe the sequence's growth rate and form relationships between its terms.

Collinear points refer to a set of points that lie on the same straight line. This concept is fundamental in coordinate geometry and a variety of mathematical theorems and proofs. To determine if points are collinear, we can examine the coordinates of these points and apply various methods.

One effective approach is to calculate the slopes between pairs of points. If the slopes are equal, this indicates that the points lie along the same line, hence they are collinear. Alternatively, if their slopes differ, the points do not lie on a single straight line, and therefore, they are non-collinear.

Collinearity is significant because it speaks to the alignment and spatial relationship of points in geometry. In the context of our problem, by showing that the given points have equal slopes, we can conclusively prove that they are collinear, meaning they lie on the same straight line.

The slope of a line in the coordinate plane expresses how steep the line is and the direction it inclines relative to the horizontal axis. It is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line, often expressed as \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

A positive slope means the line inclines upwards to the right, while a negative slope indicates it inclines downwards to the right. A zero slope means the line is horizontal, and an undefined slope (which arises when the denominator is zero) means the line is vertical.

In our exercise, by finding that the slope of line segment AB is equal to the slope of line segment BC, we determine that the points A, B, and C are collinear because they share the same slope, and thus lie on a straight line. This concept not only allows us to check for collinearity but also to graph lines, understand their direction, and analyze their behavior within the coordinate system.