If x1, x2, x3 as well as y1, y2, y3 are in G. P. with the...

If $x_{1}, x_{2}, x_{3}$ as well as $y_{1}, y_{2}, y_{3}$ are in G. P. with the same common ratio, then the points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ (A) lie on a straight line (B) lie on an ellipse (C) lie on a circle (D) are vertices of a triangle

If $x_{1}, x_{2}, x_{3}$ as well as $y_{1}, y_{2}, y_{3}$ are in G. P. with the same common ratio, then the points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$
(A) lie on a straight line
(B) lie on an ellipse
(C) lie on a circle
(D) are vertices of a triangle

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Key Concepts

Common Ratio

The common ratio is the fixed number used to multiply each term in a geometric progression to obtain the next term. Understanding the common ratio is critical, as it determines the growth or decay factor in sequences and plays a key role in linking terms together in mathematical problems.

Collinearity in Coordinate Geometry

Collinearity refers to the property of points lying on the same straight line in a coordinate plane. In problems where coordinates are generated using geometric progressions with the same common ratio, the inherent multiplicative relationships can lead to linear dependence, resulting in collinearity among the points. This concept is fundamental in understanding how algebraic sequences interact with geometric arrangements.

Geometric Progression

A geometric progression is a sequence in which each term is generated by multiplying the previous term by a constant called the common ratio. This concept is important in algebra and analysis as it lays the foundation for understanding exponential relationships and the behavior of series.

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