Question

If (1/x) + (1/y) + (1/z) = 1 and x + y + z = k, then find the value of the expression (x² + y² + z²)/(k² – 2xyz)?
1. 1
2. 0
3. 3
4. 2

Answer 1

Correct Answer - Option 1 : 1

Given:

(1/x) + (1/y) + (1/z) = 1

x + y + z = k

Formula used:

(x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

Calculation:

∵ (1/x) + (1/y) + (1/z) = 1

⇒ (xy + yz + zx)/xyz = 1

⇒ (xy + yz + zx) = xyz      -----(1)

∵ (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

⇒ (k)² = x² + y² + z² + 2(xyz)     -----(From 1)

⇒ (k)² – 2xyz = x² + y² + z²      -----(2)

∴ (x² + y² + z²)/(k² – 2xyz) = (k² – 2xyz)/(k² – 2xyz)

= 1