Question

If \(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\) and \(\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3}\) both are in GP with the same common ratio, then the points \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)\) and \(\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right)\) (a) lie on a straight line (b) lie on a ellipse (c) lie on a circle (d) are vertices of a triangle

Short answer

The points \((x_{1}, y_{1})\), \((x_{2}, y_{2})\), and \((x_{3}, y_{3})\) are vertices of a triangle.

Step-by-step solution

Express coordinates with geometric progression property

Let the common ratio of two geometric progressions be \(r\). The x coordinates, \(x_{1},\) \(x_{2}\), and \(x_{3}\), will then be a, ar, ar^2 where a is the first term in the GP. Similarly, the y coordinates, \(y_{1},\) \(y_{2}\), and \(y_{3}\), will be b, br, br^2 where b is the first term of the other GP. The given points will then be P1 \(( a, b)\), P2 \(( ar, br)\), and P3 \((ar^2, br^2)\).

Check if the points lie on a straight line

Three points are collinear (lie on a straight line) if the area of the triangle formed by them is 0. We will use the determinant method to find the area of the triangle and check if it's equal to 0. Let the points be A \((x_{A}, y_{A})\), B \((x_{B}, y_{B})\), and C \((x_{C}, y_{C})\). The area of the triangle is given by: \[Area = \dfrac{1}{2}|x_{A}(y_{B}-y_{C}) + x_{B}(y_{C}-y_{A}) + x_{C}(y_{A}-y_{B})|\] Plugging in our given points, we get: \[Area = \dfrac{1}{2}|a(br^2 - br) + ar(br^2 - b) + ar^2(b - br)|\]

Simplify the expression for the area

Now we'll simplify the expression for the area: Area \(= \dfrac{1}{2}|a\cdot b\cdot r^2 - a\cdot b\cdot r + a\cdot b\cdot r^2 - a\cdot b + a\cdot b\cdot r^2 - a\cdot b\cdot r\cdot r|\) Area \(= \dfrac{1}{2}|a\cdot b\cdot r^2 - a\cdot b\cdot r + a\cdot b\cdot r^2 - a\cdot b + a\cdot b\cdot r^2 - a\cdot b\cdot r^3|\) Now, we'll factor out the \(a\cdot b\) term: Area \(= \dfrac{1}{2} |ab \cdot (r^2 - r + r^2 - 1 + r^2 - r^3)|\) Area \(= \dfrac{1}{2} |ab \cdot (r^2 + r^2 + r^2 - r - r^3 - r - r^3 + 1)|\) Area \(= \dfrac{1}{2} |ab \cdot (3r^2 - 2r - 2r^3 + 1)|\)

Determine the shape based on the area

The area of the triangle appears to be non-zero as the expression inside the absolute value brackets depends on the common ratio r. This means that these three points do not lie on a straight line and form a triangle. Hence, we can choose option (d) as the correct answer. So, the points \((x_{1}, y_{1})\), \((x_{2}, y_{2})\), and \((x_{3}, y_{3})\) are vertices of a triangle.

Key concepts

Collinearity

When three points are said to be collinear, they all lie on a single straight line. This means that the area formed by these three points as a triangle would be zero. Collinearity often serves as a fundamental concept in geometry and is useful for solving many mathematical theorems and problems.
The role of collinearity can be seen in situations where you have to determine if multiple points lie on the same straight line based on their coordinates. If you can prove that no area forms between these points, then they are collinear.

In terms of geometric progressions, collinearity becomes a fascinating topic because if the coordinates of points are derived from a geometric progression, they are systematically related. Checking their collinearity involves special mathematical techniques such as the determinant method.

Determinant Method

The determinant method is a powerful tool in linear algebra used to find the area of a triangle formed by three points in a coordinate plane. It relies on calculating a determinant, which provides a scalar value often associated with the volume or area or the measurement of scale change.
When using the determinant to check for collinearity, the matrix is formed using the coordinates of the points. It typically takes the form:

• Row 1: \(x_1, y_1, 1\)
• Row 2: \(x_2, y_2, 1\)
• Row 3: \(x_3, y_3, 1\)

If the determinant of this matrix is zero, the points are collinear because they form no triangular area.
Calculating the determinant helps understand the geometric properties of points in space and provides a systematic way to confirm the arrangement or alignment of points—whether they form a triangle or overlap on a straight line.

Area of Triangle

The area of a triangle in a coordinate plane is determined by calculating half the absolute value of the expression known as the cross-product or determinant involving the coordinates.
The formula to calculate this area using the system of points \(x_A, y_A\), \(x_B, y_B\), and \(x_C, y_C\) is expressed as:
\[ ext{Area} = \frac{1}{2} |x_A(y_B-y_C) + x_B(y_C-y_A) + x_C(y_A-y_B)| \]
This geometric rule allows us to understand if the points form an actual triangle with a measurable area or if they align on a straight line, thus forming zero areas.
In the context of geometric progressions, understanding the area with reference to specific coordinates helps to analyze their positional attribution. If they result in a non-zero area, as in our example with common ratio \(r\), the points effectively form the vertices of a triangle rather than a straight line.