The harmonic mean between two numbers is 21/5, their A.M. `' A '` and G.M. `' G '` satisfy the relation `3A+G^2=36.` Then find the sum of square of numbers.

Explanation

To find the sum of the squares of two numbers given their harmonic mean and the relationship between their arithmetic mean and geometric mean, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- The harmonic mean (H) of two numbers \( a \) and \( b \) is \( \frac{21}{5} \).
- The relationship between their arithmetic mean (A) and geometric mean (G): \( 3A + G^2 = 36 \).

### Step 2: Write the Formulas for A, G, and H
The formulas for the means are:
- Arithmetic Mean (A):
\[ A = \frac{a + b}{2} \]
- Geometric Mean (G):
\[ G = \sqrt{ab} \]
- Harmonic Mean (H):
\[ H = \frac{2ab}{a + b} \]

### Step 3: Use the Harmonic Mean to Relate a and b
From the harmonic mean:
\[ H = \frac{2ab}{a + b} = \frac{21}{5} \]
Substituting \( H \) into the equation gives:
\[ \frac{2ab}{a + b} = \frac{21}{5} \]
Cross-multiplying yields:
\[ 10ab = 21(a + b) \]

### Step 4: Express a + b and ab
Let \( s = a + b \) and \( p = ab \). Then from the previous equation:
\[ 10p = 21s \quad \Rightarrow \quad p = \frac{21s}{10} \]

### Step 5: Substitute into the Relationship of A and G
Now, substituting \( A \) and \( G \) into the equation \( 3A + G^2 = 36 \):
- We already have \( A = \frac{s}{2} \) and \( G = \sqrt{p} \).
- Thus, \( G^2 = p \).

Substituting these into the equation:
\[ 3\left(\frac{s}{2}\right) + p = 36 \]
This simplifies to:
\[ \frac{3s}{2} + p = 36 \]

### Step 6: Substitute p in terms of s
Now substitute \( p = \frac{21s}{10} \):
\[ \frac{3s}{2} + \frac{21s}{10} = 36 \]

### Step 7: Solve for s
To solve for \( s \), we need a common denominator:
\[ \frac{15s}{10} + \frac{21s}{10} = 36 \]
Combining the fractions gives:
\[ \frac{36s}{10} = 36 \]
Multiplying both sides by 10:
\[ 36s = 360 \quad \Rightarrow \quad s = 10 \]

### Step 8: Find p
Now substitute \( s = 10 \) back to find \( p \):
\[ p = \frac{21 \times 10}{10} = 21 \]

### Step 9: Find the Sum of Squares
The sum of the squares of the numbers \( a \) and \( b \) can be found using the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting the known values:
\[ a^2 + b^2 = s^2 - 2p = 10^2 - 2 \times 21 = 100 - 42 = 58 \]

### Final Answer
The sum of the squares of the numbers is:
\[ \boxed{58} \]