If x1, x2, x3 are in GP as as y1, y2, y3 with the same common ratio, then points A(x1, y1), B(x2

step 1 In this problem, let the value of x1 is equal to a, the value of x2 is equal to a r, the value o

If x1, x2, x3 are in GP as as y1, y2, y3 with the same...

If $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}$ are in $\mathrm{GP}$ as as $\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3}$ with the same common ratio, then points $\mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right), \mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right), \mathrm{C}\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right)$ (a) satisfies $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=\mathrm{y}_{1}+\mathrm{y}_{2}+\mathrm{y}_{3}$ (b) form an equilateral triangle (c) are collinear (d) form an isosceles triangle

If $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}$ are in $\mathrm{GP}$ as as $\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3}$ with the same common ratio, then points $\mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right), \mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right), \mathrm{C}\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right)$
(a) satisfies $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=\mathrm{y}_{1}+\mathrm{y}_{2}+\mathrm{y}_{3}$
(b) form an equilateral triangle
(c) are collinear
(d) form an isosceles triangle

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Key Concepts

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed non-zero constant called the common ratio. This concept is essential when relating the coordinates of points since the multiplicative relationships among the terms can lead to proportional relationships between coordinate pairs.

Common Ratio

The common ratio is the constant factor between consecutive terms in a geometric progression. Its presence in both the x-coordinates and y-coordinates provides a link between the two sets, establishing multiplicative relationships that can affect the geometric arrangement of the corresponding points.

Collinearity in Coordinate Geometry

Collinearity refers to the property of points lying on the same straight line. In coordinate geometry, points are collinear if the slopes between any two pairs of points are equal or if the area of the triangle formed by the points is zero. Establishing collinearity often involves using the relationships derived from sequences such as geometric progressions.

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