Question

If a² + b² + c² = 90, ab + bc + ca = 83 and a³ + b³ + c³ = 532, Find the value of (1/a + 1/b + 1/c)?
1. 73/150
2. 83/140
3. 93/160
4. 87/140

Answer 1

Correct Answer - Option 2 : 83/140

Given:

a2 + b2 + c2 = 90

ab + bc + ca = 83

a3 + b3 + c3 = 532

Formula used:

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

a³ + b³ + c³ - 3abc = (a + b + c){a2 + b2 + c2 - (ab + bc + ca)}

Calculation:

First simplify what is asked

⇒ (1/a + 1/b + 1/c)

⇒ (ab + bc + ca)/abc      ----(1)

Now, find (a + b + c)

⇒ (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

⇒ (a + b +c)² = 90 + 2 × 83

⇒ (a + b +c)2 = 256

⇒ (a + b + c) = 16

Put value in 2nd formula

⇒ a3 + b3 + c3 - 3abc = (a + b + c){a2 + b2 + c2 - (ab + bc + ca)}

⇒ 532 - 3abc = (16)(90 - 83)

⇒ 3abc = 532 - 112

⇒ 3abc = 420

⇒ abc = 140

Put the values in the equation (1)

⇒ 83/140

∴ The value of (1/a + 1/b + 1/c) is 83/140.