If `(a+b+c)^(2)=36,ab+bc+ca=11 and a,b,c in N`, then find `a^(2)+b^(2)+c^(2)`.

Options

A. 17
B. 22
C. 14
D. 6

Answer

To solve the problem step by step, we will use the given information and the formula for the square of a sum.

### Step 1: Write down the given equations.
We have:
1. \((a + b + c)^2 = 36\)
2. \(ab + bc + ca = 11\)

### Step 2: Use the identity for the square of a sum.
We know that:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

### Step 3: Substitute the known values into the identity.
From the first equation, we have:
\[ 36 = a^2 + b^2 + c^2 + 2(11) \]

### Step 4: Simplify the equation.
Calculating \(2(11)\):
\[ 2(11) = 22 \]
Now substitute this back into the equation:
\[ 36 = a^2 + b^2 + c^2 + 22 \]

### Step 5: Isolate \(a^2 + b^2 + c^2\).
To find \(a^2 + b^2 + c^2\), we rearrange the equation:
\[ a^2 + b^2 + c^2 = 36 - 22 \]

### Step 6: Perform the subtraction.
Calculating \(36 - 22\):
\[ a^2 + b^2 + c^2 = 14 \]

### Final Answer:
Thus, the value of \(a^2 + b^2 + c^2\) is \(14\).

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