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

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

The correct answer is (D) \(x^2 - 18x + 16 = 0\).

Step 1: Understand the Problem

We have two numbers with an arithmetic mean of 9 and a geometric mean of 4. We need to find the equation for which these numbers are the roots.

Step 2: Define Arithmetic and Geometric Mean

The arithmetic mean (AM) of two numbers, say, \(a\) and \(b\), is given by \(\frac{a + b}{2} = 9\). The geometric mean (GM) is given by \(\sqrt{ab} = 4\).

Step 3: Set Up Equations from Means

1. From the arithmetic mean: \(a + b = 18\).2. From the geometric mean: \(ab = 16\).

Step 4: Derive the Quadratic Equation

Using the properties of roots of a quadratic equation, where for quadratic \(x^2 - (sum\ of\ the\ roots)\cdot x + (product\ of\ the\ roots) = 0\), substitute the sums and products:Given: \(a + b = 18\), \(ab = 16\)Thus, the quadratic equation becomes: \(x^2 - 18x + 16 = 0\).

Step 5: Verify with Options

We compare the derived quadratic equation \(x^2 - 18x + 16 = 0\) with the options given:- Option (A) is \(x^2 + 18x + 16 = 0\)- Option (B) is \(x^2 - 18x - 16 = 0\)- Option (C) is \(x^2 + 18x - 16 = 0\)- Option (D) is \(x^2 - 18x + 16 = 0\)Therefore, Option (D) is correct.

Key Concepts

Arithmetic Mean

To find the arithmetic mean (AM) of two numbers, you add them together and divide the result by 2. This is a straightforward method to find what is essentially the average of two values.
In our problem, the arithmetic mean is given as 9. Let the two numbers be \(a\) and \(b\). Thus, the equation for the AM is:

• \(\frac{a + b}{2} = 9\)

This results in the equation \(a + b = 18\). Understanding the arithmetic mean is key in forming the sum equation when solving problems involving quadratic equations.

Geometric Mean

The geometric mean (GM) is a measure that indicates the central tendency of two numbers by using the product of these numbers. Unlike the arithmetic mean, which adds, the geometric mean multiplies.
In this context, the GM is used to satisfy \(\sqrt{ab} = 4\). This can be translated to:

• \(ab = 16\)

This equation represents the product of the two numbers. The geometric mean gives us a way to equate the product, which is crucial when dealing with equations that require both sum and product of numbers, such as quadratic equations.

Roots of Equations

The roots of a quadratic equation are the solutions for the variable \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). In this exercise, the two numbers derived from the AM and GM are the roots of the equation.
The properties of polynomial equations tell us that these roots are related to the coefficients of the quadratic polynomial. Particularly:

• The sum of the roots \((a + b)\) is equal to the negative of the linear coefficient \(b\)
• The product of the roots \((ab)\) is equal to the constant term \(c\)

This understanding allows us to derive the quadratic equation based on given or calculated roots, ensuring we solve for x accurately.

Properties of Quadratic Equations

Quadratic equations present some vital mathematical properties useful for solving diverse problems. These properties revolve around the sum and product of their roots. Here is a quick look at their significance in mathematics:

• The quadratic equation can be expressed as: \[x^2 -(\text{sum of the roots})\cdot x + (\text{product of the roots}) = 0\]
• The sum of the roots \((a+b)\) is derived from the coefficient of \(x\)
• The product of the roots \(ab\) relates to the constant term of the equation

In our problem, knowing \(a + b = 18\) and \(ab = 16\), we can tailor the quadratic equation \(x^2 - 18x + 16 = 0\). Understanding these properties allows students to see the direct connection between given values and forming a quadratic equation.