Problem 18 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c\) form an A. P. with common difference \(d(\neq 0)\) and \(x, y, z\) form a G. P. with common ratio \(r \neq 1\) ), then the area of the triangle with vertices \((a, x),(b, y)\) and \((c, z)\) is independent of (A) \(b\) (B) \(r\) (C) \(d\) (D) \(x\)

Step-by-Step Solution

Answer

The area is independent of (B) \(r\).

Step 1: Understand Arithmetic Progression (A.P.) and Geometric Progression (G.P.)

From the problem, we know that the terms \(a, b,\) and \(c\) form an arithmetic progression (A.P.) which implies:\[b = a + d, \quad c = a + 2d\] where \(d\) is the common difference.Additionally, \(x, y,\) and \(z\) form a geometric progression (G.P.) which implies:\[y = xr, \quad z = xr^2\] where \(r\) is the common ratio.

Step 2: Use Area Formula for the Triangle

The area of a triangle given vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is given by:\[\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]Substitute the vertices \((a, x), (b, y), (c, z)\):\[\text{Area} = \frac{1}{2} \left| a(y-z) + b(z-x) + c(x-y) \right|\]

Step 3: Substitute A.P. and G.P. Values

Substitute \(b = a + d\), \(c = a + 2d\), \(y = xr\), and \(z = xr^2\):\[\text{Area} = \frac{1}{2} \left| a(xr-xr^2) + (a + d)(xr^2-x) + (a + 2d)(x-xr) \right|\] This simplifies to:\[\text{Area} = \frac{1}{2} \left| axr(1-r) - dxr^2 + dx + ax(1-r) + 2dx - 2dxr \right|\]

Step 4: Simplify the Expression

Simplify the expression by grouping like terms:\[\text{Area} = \frac{1}{2} \left| ax(1-r)(1-r) + dx(1 - r^2) \right|\]Factor out \(x(1-r)\):\[\text{Area} = \frac{1}{2} \left| x(1-r)( a(1-r) + d(1 + r) ) \right|\]

Step 5: Determine Independence

Notice that the expression for the area only contains \(a, d, x, r\), the constants.However, \(b\) solely depends on \(a\) and \(d\). The area formula's structure indicates some dependencies, but crucially, it is independent of a change in \(r\) since it forms merely a constant ratio in each of the calculation terms after factoring.

Key Concepts

Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers in which each term after the first is obtained by adding a constant difference, called the common difference. This pattern is predictable and allows us to easily determine subsequent terms in the sequence.

A common formula used for A.P. terms is:

• Second term: \( b = a + d \)
• Third term: \( c = a + 2d \)

This mathematical arrangement is essential in various applications, especially when needing to express relationships that have a consistent interval, such as time sequences or distances.

Understanding A.P. helps in solving problems as it creates a straightforward framework for evaluating relationships between numbers.

Geometric Progression

In contrast to an A.P., a Geometric Progression (G.P.) is a sequence where each term after the first is obtained by multiplying the previous term by a fixed and non-zero number called the common ratio. The multiplication factor, which can be any real number except zero, drives the sequence.

Important G.P. terms are formulated as:

• Second term: \( y = xr \)
• Third term: \( z = xr^2 \)

The exponentially growing or decreasing nature of G.P. finds use in contexts where exponential change occurs, such as population growth or compound interest.

Recognizing the growth pattern in a G.P. is crucial for understanding and predicting values within these series efficiently.

Common Difference

In an A.P., the common difference (\(d\)) is the fixed amount that each term increases or decreases by, as you move from one term to the next. It serves as the key characteristic distinguishing one arithmetic sequence from another.

The role of the common difference is evident in transforming the arithmetic sequence:

• From the first term \( a \) to the second term: \( b = a + d \)
• From the second term to the third term: \( c = a + 2d \)

Even small variations in the common difference can lead to entirely distinct sequences over time.

Recognizing and manipulating the common difference helps solve problems regarding sequence formulation and projections.

Common Ratio

The common ratio (\(r\)) in a G.P. defines the relationship between consecutive terms, signifying the factor by which a term is multiplied to obtain the next term. Unlike the linear shift seen in A.P., this ratio involves multiplication, leading to exponential changes.

For instance, a G.P. with terms \(x, y, z\) has:

• \(y = xr\)
• \(z = xr^2\)

Understanding the common ratio is vital when dealing with phenomena demonstrating exponential growth or decay, such as economic models or geometric fractals.

The ability to identify and utilize the common ratio effectively is key to solving progression-related mathematical problems.