Question

If \[ \frac{1}{a(b + c)} + \frac{1}{b(c + a)} + \frac{1}{c(a + b)} = k, \text{ then the value of } k \text{ is:} \]

• A. \( ab + bc + ca \)
• B. \( (ab + bc + ca)^2 \)
• C. \( 2(ab + bc + ca) \)
• D. 0

Hint:

When simplifying rational expressions with multiple terms, look for symmetry and common denominators to combine terms.

Answer 1

The Correct Option is C

Step 1: Simplify the equation.
Combine the fractions: \[ \frac{1}{a(b + c)} + \frac{1}{b(c + a)} + \frac{1}{c(a + b)} = \frac{ab + bc + ca}{(a + b)(b + c)(c + a)} \]

Step 2: Match the result.
The simplified equation yields \( k = 2(ab + bc + ca) \). Therefore, the correct answer is 3. \( 2(ab + bc + ca) \).