Problem 88 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(x_{1}, x_{2}, x_{3}\) and \(y_{1}, y_{2}, y_{3}\) are both 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) are vertices of a triangle (b) lie on a straight line (c) lie on an ellipse (d) lie on a circle.

Step-by-Step Solution

Answer

The points lie on a straight line (option b).

Step 1: Understand the Problem

We need to determine the geometric relationship of the points \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) given that both sets \(x_1, x_2, x_3\) and \(y_1, y_2, y_3\) form a geometric progression (G.P.) with the same common ratio.

Step 2: Define Geometric Progression

A sequence is in geometric progression if each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Let's assume the common ratio is \(r\). Thus, the sequences can be defined as \(x_2 = x_1 \cdot r\), \(x_3 = x_1 \cdot r^2\) for \(x\) values and similarly \(y_2 = y_1 \cdot r\), \(y_3 = y_1 \cdot r^2\) for \(y\) values.

Step 3: Express Each Point in Terms of Common Ratio

The points can be expressed as: \((x_1, y_1), (x_1 \cdot r, y_1 \cdot r), (x_1 \cdot r^2, y_1 \cdot r^2)\). This representation shows how each following point is a scaled version of the initial point \((x_1, y_1)\) by the common ratio \(r\).

Step 4: Analyze Collinearity Condition

For the points to lie on a straight line, the determinant of the matrix formed by these points must be zero:\[\begin{vmatrix}x_1 & y_1 & 1 \x_2 & y_2 & 1 \x_3 & y_3 & 1 \\end{vmatrix} = 0\]Substituting the values, we get:\[\begin{vmatrix}x_1 & y_1 & 1 \x_1 \cdot r & y_1 \cdot r & 1 \x_1 \cdot r^2 & y_1 \cdot r^2 & 1 \\end{vmatrix} = 0\]

Step 5: Evaluate the Determinant

The determinant for this setup is zero as each (2nd, 3rd points) is a scalar multiple of the previous, making the rows linearly dependent. Thus,\[\begin{vmatrix}x_1 & y_1 & 1 \x_1 \cdot r & y_1 \cdot r & 1 \x_1 \cdot r^2 & y_1 \cdot r^2 & 1 \\end{vmatrix} = 0\]This confirms the points are collinear.

Key Concepts

Collinearity

Collinearity is a geometric concept where points lie on a single straight line. In this context, determining if three points are collinear is essential to identifying their geometric arrangement. When this happens, the points do not form a triangle, but rather, they maintain alignment in a linear fashion.
To test for collinearity, we can use the determinant of a matrix approach. Here, if the determinant of the matrix composed of these points equals zero, it indicates that the points are collinear. This is because a zero determinant reveals that the rows of the matrix are linearly dependent, hence they fall on a straight line.
In our problem, substituting the geometric progression terms into the determinant matrix shows that the determinant is indeed zero, confirming collinearity.

Determinants

Determinants are a helpful mathematical tool used primarily in linear algebra to determine if a set of vectors is linearly independent, among other applications. In the context of collinearity, using determinants is a quick way to prove alignment of points.
The determinant is calculated from a square matrix. For collinearity involving 2D coordinates, we use the matrix comprising three points extended by a column of ones: \[ \begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{vmatrix} \]By evaluating this determinant, a zero result indicates that the points lay in a straight line because the rows are linearly dependent. Thus, the properties of determinants simplify recognizing collinear points.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It bridges algebra and geometry, allowing the graphical representation of algebraic equations.
In coordinate geometry, points are described as coordinates, introducing a numerical perspective to geometry. This enables us to apply algebraic methods, like calculating distances, midpoints, and checking collinearity through determinants, to geometric figures.
The use of geometric progressions where points and their respective coordinates have a common ratio adds another layer of structure to problems like this, where the coordinates themselves reflect a consistent arithmetic relationship. Essentially, coordinate geometry provides the tools to translate geometric scenarios into algebraic calculations, making complex spatial problems easier to solve.