Problem 15 - A Complete Resource Book in Mathematics for JEE Main

Question

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

Step-by-Step Solution

Answer

(A) lie on a straight line

Step 1: Understanding Geometric Progression (G.P.)

Given that \(x_1, x_2, x_3\) and \(y_1, y_2, y_3\) are in a geometric progression with a common ratio \(r\), we have:\[ x_2 = x_1 \cdot r, \quad x_3 = x_1 \cdot r^2 \]\[ y_2 = y_1 \cdot r, \quad y_3 = y_1 \cdot r^2 \]

Step 2: Analyzing Collinearity Condition

To determine if the points \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) lie on the same straight line, we can use the condition for collinearity:\[ (x_2 - x_1)(y_3 - y_1) = (y_2 - y_1)(x_3 - x_1) \]

Step 3: Applying Collinearity Condition

Substitute the expressions for \(x_2, x_3, y_2,\) and \(y_3\) in terms of \(x_1, y_1,\) and \(r\):\[(x_1 \cdot r - x_1)(y_1 \cdot r^2 - y_1) = (y_1 \cdot r - y_1)(x_1 \cdot r^2 - x_1)\]Simplifying both sides:\[x_1(y_1 r^2 - y_1)r = y_1(x_1 r^2 - x_1)r\]Simplify further to check equality, which holds true, implying that both sides are equal.

Step 4: Conclusion on Collinearity

Since the condition for collinearity holds true, the points \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) lie on a straight line.

Key Concepts

Geometric Progression

A Geometric Progression (G.P.) 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 is a central concept in understanding how sequences unfold based on patterns of multiplication.

For example, if we have a sequence starting with a term \(x_1\), and a common ratio \(r\), then:

• The second term, \(x_2\), is \(x_1 \times r\)
• The third term, \(x_3\), is \(x_1 \times r^2\)

The same pattern applies to another sequence starting with \(y_1\).

This consistent multiplication by the common ratio gives the sequence a predictable pattern, and it is what makes geometric progression so powerful in mathematics. Not only does it allow for understanding growth patterns, but it also sets the stage for examining relationships between terms, especially in coordinate systems.

Collinearity

Collinearity is the condition where three or more points lie on a single straight line. To determine if points like \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) are collinear, there is a simple algebraic condition that needs to be satisfied.

The mathematical check for collinearity among these points involves ensuring that the area formed by a triangle with these points as vertices is zero. The equivalent algebraic form of this condition is:\[(x_2 - x_1)(y_3 - y_1) = (y_2 - y_1)(x_3 - x_1)\]This equation basically checks the slope between the first two points and the slope between the last two points. If the slopes are equal, all points lie on the same line.

Using this condition in coordinate geometry, we can easily verify the alignment of points and subsequently draw conclusions about their spatial relationship.

Common Ratio

The common ratio in a geometric progression is a key component that influences the spacing and direction of points in a sequence. It is the constant factor by which each term of a sequence is multiplied to get the next term.

In the original problem, the common ratio \(r\) is the same for both the sequences \((x_1, x_2, x_3)\) and \((y_1, y_2, y_3)\). This uniform common ratio plays a critical role in ensuring that the sequences grow consistently, maintaining their relative proportions.

This factor, \(r\), ensures that the points generated from these sequences can potentially align in symmetrical ways, such as forming straight lines when examined in a coordinate system. By manipulating or understanding \(r\), one can predict the expansion or contraction of a sequence in a geometric setting.