Problem 89 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let two numbers have arithmetic mean 9 and geometric mean 4 . Then these numbers are the roots of the quadratic equation (a) \(x^{2}-18 x-16=0\) (b) \(x^{2}-18 x+16=0\) (c) \(x^{2}+18 x-16=0\) (d) \(x^{2}+18 x+16=0\)

Step-by-Step Solution

Answer

Option (b), \(x^2 - 18x + 16 = 0\).

Step 1: Understand the Arithmetic Mean

The arithmetic mean of two numbers, say \(a\) and \(b\), is given as \(\frac{a+b}{2} = 9\). This simplifies to the equation \(a+b = 18\).

Step 2: Understand the Geometric Mean

The geometric mean of two numbers, \(a\) and \(b\), is given as \(\sqrt{ab} = 4\). Squaring both sides, we find \(ab = 16\).

Step 3: Formulate the Quadratic Equation

The numbers in question are the roots of the quadratic equation \(x^2 - (a+b)x + ab = 0\). Substituting the values from the previous steps, we have \(x^2 - 18x + 16 = 0\).

Step 4: Match with Given Options

Comparing the obtained equation \(x^2 - 18x + 16 = 0\) with the given options, we find that it matches option (b).

Key Concepts

Arithmetic Mean

When we talk about the arithmetic mean, we mean the simplest form of averaging numbers. The arithmetic mean of two numbers, say \(a\) and \(b\), is found by adding them together and then dividing by two. This is expressed through the formula \( \frac{a+b}{2} \). In the original exercise, the arithmetic mean is given as 9. This means that \( \frac{a+b}{2} = 9 \), which simplifies to the equation \( a+b = 18 \).
This average is useful in many everyday situations. It's a basic measure of central tendency, giving us a single value that represents the center of a set of numbers. In the context of quadratic equations, this mean helps in finding the sum of the roots when they are unknown quantities. The sum (\( a+b\)) here contributes directly to forming the quadratic equation.

Geometric Mean

The geometric mean involves a different kind of averaging process. It's calculated by multiplying the two numbers and then taking the square root. This is written as \( \sqrt{ab} \). In our exercise, the geometric mean is given as 4, which gives us \( \sqrt{ab} = 4 \). Squaring both sides, we find \( ab = 16 \).
This means that if we have two numbers, their multiplication results in 16. The geometric mean tends to be more appropriate in growth rates, scales, or when dealing with percentage increases (like interest rates).
In terms of quadratic equations, the product \( ab \) directly informs the constant term in the equation, which is crucial for finding the polynomial's roots. It's a handy mean to use, particularly when dealing with ratios or exponential data.

Roots of a Polynomial

Roots of a polynomial are the values that make the polynomial equal to zero. For a quadratic polynomial like \( ax^2 + bx + c = 0 \), these are the solutions \( x \) that satisfy the equation. In our problem, the numbers corresponding to the arithmetic and geometric means are the roots of this quadratic equation.

• The sum of the roots can be calculated using \( -(b/a) \), and here it is \( 18 \), as given by \( a+b = 18 \).
• The product of the roots is \( c/a \), thus giving us \( ab = 16 \).

Substituting these values into the standard quadratic form \( x^2 - (a+b)x + ab = 0 \), we arrive at \( x^2 - 18x + 16 = 0 \).
Understanding the roots and their relationship with the coefficients of the polynomial provides a full picture of how the numbers \( a \) and \( b \) fit within the equation. This insight is fundamental for solving quadratic equations and predicting their properties.