If `(x)/(a)=(y)/(b)=(z)/(c)` then `xy+yz+zx` is equal to :

Options

A. `((a+b+c)^(2))/(x^(2)+y^(2)+z^(2))`
B. `(x^(2)(a+b+c)^(2)-a^(2)(x^(2)+y^(2)+z^(2)))/(2a^(2))`
C. `(ax+by+cz)/((a+b+c)^(2))`
D. none of the abve

Answer

To solve the equation \(\frac{x}{a} = \frac{y}{b} = \frac{z}{c}\) and find the value of \(xy + yz + zx\), we can follow these steps:

### Step 1: Set a common ratio
Let us denote the common ratio as \(k\). Therefore, we can express \(x\), \(y\), and \(z\) in terms of \(k\):
\[ x = ak, \quad y = bk, \quad z = ck \]

### Step 2: Substitute the expressions into the equation
Now, we substitute these expressions into the equation \(xy + yz + zx\):
\[ xy + yz + zx = (ak)(bk) + (bk)(ck) + (ck)(ak) \]

### Step 3: Simplify each term
Calculating each term:
\[ xy = abk^2, \quad yz = bck^2, \quad zx = ack^2 \]
So, we can write:
\[ xy + yz + zx = abk^2 + bck^2 + ack^2 \]

### Step 4: Factor out the common term
Now, we can factor out \(k^2\):
\[ xy + yz + zx = k^2(ab + bc + ac) \]

### Step 5: Conclusion
Thus, we have expressed \(xy + yz + zx\) in terms of \(k\):
\[ xy + yz + zx = k^2(ab + bc + ac) \]

### Final Result
The expression \(xy + yz + zx\) is equal to \(k^2(ab + bc + ac)\).