If `a + b + c = 4 sqrt(3) and a^(2) + b^(2) + c^(2)= 16` then the ratio a : b : c is

Options

A. `1 : 1 : 1 `
B. ` 1 : sqrt(2) : sqrt(3)`
C. ` 1 : 2 : 3 `
D. None of these

Answer

To solve the problem, we start with the given equations:

1. \( a + b + c = 4\sqrt{3} \)
2. \( a^2 + b^2 + c^2 = 16 \)

We need to find the ratio \( a : b : c \).

### Step 1: Use the identity for the square of sums

We know that:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

### Step 2: Substitute the known values

Substituting the known values into the equation:

\[ (4\sqrt{3})^2 = 16 + 2(ab + bc + ca) \]

Calculating \( (4\sqrt{3})^2 \):

\[ 48 = 16 + 2(ab + bc + ca) \]

### Step 3: Isolate \( ab + bc + ca \)

Now, we can isolate \( ab + bc + ca \):

\[ 48 - 16 = 2(ab + bc + ca) \]

\[ 32 = 2(ab + bc + ca) \]

\[ ab + bc + ca = 16 \]

### Step 4: Set up a system of equations

Now we have the following equations:

1. \( a + b + c = 4\sqrt{3} \)
2. \( a^2 + b^2 + c^2 = 16 \)
3. \( ab + bc + ca = 16 \)

### Step 5: Use the equations to find the ratio

Let’s assume \( a, b, c \) can be expressed in terms of a common variable \( k \):

Let \( a = kx, b = ky, c = kz \) where \( x + y + z = 4\sqrt{3}/k \) and \( x^2 + y^2 + z^2 = 16/k^2 \).

From \( ab + bc + ca = 16 \):

\[ k^2(xy + yz + zx) = 16 \]

### Step 6: Assume equal values for simplicity

Assuming \( a = b = c \):

Let \( a = b = c = x \).

Then we have:

\[ 3x = 4\sqrt{3} \implies x = \frac{4\sqrt{3}}{3} \]

Now substituting \( x \) back into the second equation:

\[ 3x^2 = 16 \implies x^2 = \frac{16}{3} \implies x = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \]

### Step 7: Find the ratio

Thus, we have:

\[ a = b = c = \frac{4\sqrt{3}}{3} \]

So the ratio \( a : b : c = 1 : 1 : 1 \).

### Final Answer

The ratio \( a : b : c \) is \( 1 : 1 : 1 \).

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