If Arithmetic mean and Geometric mean of two positive numbers a and b are 10 and 8 respectively, find the numbers:

Options

A. 2, 4
B. 4, 16
C. 4, 8
D. 2, 16

Answer

To solve the problem, we need to find two positive numbers \( a \) and \( b \) given that their Arithmetic Mean (AM) is 10 and their Geometric Mean (GM) is 8.

### Step-by-Step Solution

1. **Understanding the Arithmetic Mean (AM)**:
The formula for the Arithmetic Mean of two numbers \( a \) and \( b \) is:
\[ AM = \frac{a + b}{2} \]
Given that \( AM = 10 \), we can set up the equation:
\[ \frac{a + b}{2} = 10 \]
Multiplying both sides by 2 gives:
\[ a + b = 20 \quad \text{(Equation 1)} \]

2. **Understanding the Geometric Mean (GM)**:
The formula for the Geometric Mean of two numbers \( a \) and \( b \) is:
\[ GM = \sqrt{ab} \]
Given that \( GM = 8 \), we can set up the equation:
\[ \sqrt{ab} = 8 \]
Squaring both sides gives:
\[ ab = 64 \quad \text{(Equation 2)} \]

3. **Substituting Equation 1 into Equation 2**:
From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = 20 - a \]
Now, substitute this expression for \( b \) into Equation 2:
\[ a(20 - a) = 64 \]
Expanding this gives:
\[ 20a - a^2 = 64 \]
Rearranging the equation results in:
\[ a^2 - 20a + 64 = 0 \quad \text{(Quadratic Equation)} \]

4. **Solving the Quadratic Equation**:
We can solve the quadratic equation \( a^2 - 20a + 64 = 0 \) using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -20 \), and \( c = 64 \):
\[ a = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 1 \cdot 64}}{2 \cdot 1} \]
\[ a = \frac{20 \pm \sqrt{400 - 256}}{2} \]
\[ a = \frac{20 \pm \sqrt{144}}{2} \]
\[ a = \frac{20 \pm 12}{2} \]
This gives us two potential solutions:
\[ a = \frac{32}{2} = 16 \quad \text{and} \quad a = \frac{8}{2} = 4 \]

5. **Finding Corresponding Values of \( b \)**:
Using \( a + b = 20 \):
- If \( a = 16 \), then \( b = 20 - 16 = 4 \).
- If \( a = 4 \), then \( b = 20 - 4 = 16 \).

Thus, the two positive numbers are \( 16 \) and \( 4 \).

### Final Answer:
The two positive numbers are \( 16 \) and \( 4 \).