Problem 42 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let two numbers have arithmetic mean 9 and geometric mean 4 . Then these numbers are the roots of the quadratic equation [2004] (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

(D) \(x^{2}-18x+16=0\)

Step 1: Define the Relationships

Let the two numbers be \( a \) and \( b \). We know their arithmetic mean is \( (a+b)/2 = 9 \). Therefore, we have the equation \( a + b = 18 \).

Step 2: Use the Geometric Mean

The geometric mean given is 4, which means \( \sqrt{ab} = 4 \). Squaring both sides, we obtain \( ab = 16 \).

Step 3: Form the Quadratic Equation Using Roots

A quadratic equation with roots \( a \) and \( b \) can be written as \( x^2 - (a+b)x + ab = 0 \). Substitute the values we have: \( a + b = 18 \) and \( ab = 16 \). Thus, the equation becomes \( x^2 - 18x + 16 = 0 \).

Step 4: Match the Equation to the Options

Among the provided options, \( x^2 - 18x + 16 = 0 \) corresponds to option (D).

Key Concepts

Arithmetic Mean

The arithmetic mean is the average of a set of numbers. It is calculated by adding all the numbers together and dividing the sum by the count of the numbers. For example, if two numbers are represented as \( a \) and \( b \), their arithmetic mean is given by \((a + b)/2\). In this context, we use the arithmetic mean to find a relationship between two unknown numbers.

• In the problem, we are told that the arithmetic mean of two numbers is 9. The equation from this is \((a + b)/2 = 9\).
• Solving this equation gives us \(a + b = 18\).

Understanding the arithmetic mean helps us know how two numbers are related through their sum. This sum becomes a part of the characteristics of the quadratic equation with these numbers as roots.

Geometric Mean

The geometric mean provides another type of average that multiplies numbers and then takes the nth root, where \(n\) is the count of numbers. For two numbers, it's calculated as \(\sqrt{ab}\), where \(a\) and \(b\) are the numbers in question. It gives a different perspective of the numbers' relationship compared to the arithmetic mean.

• In the exercise, the geometric mean of our numbers is 4. This means \(\sqrt{ab} = 4\).
• Squaring both sides to remove the square root gives \(ab = 16\).

The geometric mean creates an equation relating to the product of \(a\) and \(b\), which is crucial for forming the quadratic equation. These relationships together help pin down the exact values for the numbers needed in the exercise.

Roots of Equations

The roots of a quadratic equation, often denoted as \(a\) and \(b\), are the solutions where the equation equals zero. Quadratic equations can be expressed in the form \(x^2 - (a+b)x + ab = 0\). Here, \(a + b\) and \(ab\) are derived from the arithmetic and geometric means.

• Using the information \(a + b = 18\) and \(ab = 16\), we substitute these into the standard form.
• Thus, the equation becomes \(x^2 - 18x + 16 = 0\).
• Finding this equation corresponds to one of the solutions given: option (D).

This shows how knowing the sum and product of roots allows us to construct the original equation, reaffirming their significance in solving quadratics accurately.