If xy = a, xz = b, yz = c and `abc ne 0`, find the value of `x^2 + y^2 + z^2` in terms of a, b, c.

Answer

To find the value of \( x^2 + y^2 + z^2 \) in terms of \( a, b, c \) given the equations \( xy = a \), \( xz = b \), and \( yz = c \), we can follow these steps:

### Step 1: Express \( x, y, z \) in terms of \( a, b, c \)

From the equations:
1. \( xy = a \) implies \( y = \frac{a}{x} \)
2. \( xz = b \) implies \( z = \frac{b}{x} \)
3. \( yz = c \) implies \( z = \frac{c}{y} \)

### Step 2: Substitute \( y \) and \( z \) in terms of \( x \)

Using \( y = \frac{a}{x} \) in the equation for \( z \):
\[ z = \frac{c}{y} = \frac{c}{\frac{a}{x}} = \frac{cx}{a} \]

Now we have:
- \( y = \frac{a}{x} \)
- \( z = \frac{cx}{a} \)

### Step 3: Substitute \( y \) and \( z \) into \( x^2 + y^2 + z^2 \)

Now we can express \( x^2 + y^2 + z^2 \):
\[ x^2 + y^2 + z^2 = x^2 + \left(\frac{a}{x}\right)^2 + \left(\frac{cx}{a}\right)^2 \]

### Step 4: Simplify the expression

Calculating each term:
\[ y^2 = \left(\frac{a}{x}\right)^2 = \frac{a^2}{x^2} \]
\[ z^2 = \left(\frac{cx}{a}\right)^2 = \frac{c^2x^2}{a^2} \]

Thus,
\[ x^2 + y^2 + z^2 = x^2 + \frac{a^2}{x^2} + \frac{c^2x^2}{a^2} \]

### Step 5: Combine the terms

To combine these terms, we can factor out \( x^2 \):
\[ x^2 + \frac{a^2}{x^2} + \frac{c^2x^2}{a^2} = x^2 \left(1 + \frac{c^2}{a^2}\right) + \frac{a^2}{x^2} \]

### Step 6: Use the product \( xyz \)

Now, we know that:
\[ xyz = \sqrt{(xy)(xz)(yz)} = \sqrt{abc} \]
Thus,
\[ x^2y^2z^2 = abc \implies (xyz)^2 = abc \]

### Step 7: Final expression

Now substituting \( xyz \) back into the equation:
\[ x^2 + y^2 + z^2 = \frac{b^2}{c^2} + \frac{a^2}{b^2} + \frac{a^2}{c^2} \]

### Conclusion

Thus, the final expression for \( x^2 + y^2 + z^2 \) in terms of \( a, b, c \) is:
\[ x^2 + y^2 + z^2 = \frac{b^2}{c^2} + \frac{a^2}{b^2} + \frac{a^2}{c^2} \]