If `a ^(2) + b ^(2) + c ^(2) = 16, x ^(2) + y ^(2) + z^(2) = 25 and ax + by + cz = 20 ` then the value of `(a + b + c)/( x + y + c)`

Options

A. `3/5`
B. `5/3`
C. `4/5`
D. `5/4`

Answer

To solve the problem step by step, we need to find the value of \(\frac{a + b + c}{x + y + z}\) given the equations:

1. \(a^2 + b^2 + c^2 = 16\)
2. \(x^2 + y^2 + z^2 = 25\)
3. \(ax + by + cz = 20\)

### Step 1: Analyze the first equation
From the equation \(a^2 + b^2 + c^2 = 16\), we can assume values for \(b\) and \(c\) to simplify our calculations. Let's set \(b = 0\) and \(c = 0\).

\[ a^2 + 0^2 + 0^2 = 16 \implies a^2 = 16 \implies a = 4 \quad (\text{taking the positive root}) \]

### Step 2: Analyze the second equation
Now, let's look at the second equation \(x^2 + y^2 + z^2 = 25\). Similarly, we can set \(y = 0\) and \(z = 0\).

\[ x^2 + 0^2 + 0^2 = 25 \implies x^2 = 25 \implies x = 5 \quad (\text{taking the positive root}) \]

### Step 3: Verify the third equation
Now we need to check if these values satisfy the third equation \(ax + by + cz = 20\).

Substituting the values we found:
\[ ax + by + cz = 4 \cdot 5 + 0 \cdot 0 + 0 \cdot 0 = 20 \]

This equation holds true.

### Step 4: Calculate \(a + b + c\) and \(x + y + z\)
Now we can calculate \(a + b + c\) and \(x + y + z\):

\[ a + b + c = 4 + 0 + 0 = 4 \]
\[ x + y + z = 5 + 0 + 0 = 5 \]

### Step 5: Find the final value
Now we can find the required value:

\[ \frac{a + b + c}{x + y + z} = \frac{4}{5} \]

Thus, the final answer is \(\frac{4}{5}\).

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