If ab = 21 and (a + b)^2 / (a − b)^2 = 25/4, then find the value of: a^2 + b^2 + 3ab Use algebraic identities to express everything in terms of a^2 + b^2 and ab.

Introduction / Context:
This question tests using identities for (a + b)^2 and (a − b)^2 and solving a rational equation without finding a and b individually. When a ratio like (a + b)^2/(a − b)^2 is given, you should rewrite each square in terms of S = a^2 + b^2 and the product ab. The key identities are: (a + b)^2 = a^2 + b^2 + 2ab (a − b)^2 = a^2 + b^2 − 2ab With ab known (21), you can substitute and solve for S using the ratio equation. Once S is found, the target a^2 + b^2 + 3ab becomes S + 3ab, which is immediate. This approach is standard in aptitude “simplification” problems because it avoids quadratic solving and keeps computations controlled. The main difficulty is careful cross-multiplication and not mixing up +2ab and −2ab in the identities.

Given Data / Assumptions:

• ab = 21 • (a + b)^2 / (a − b)^2 = 25/4 • Let S = a^2 + b^2 • Identities: (a + b)^2 = S + 2ab, (a − b)^2 = S − 2ab • Required: a^2 + b^2 + 3ab = S + 3ab

Concept / Approach:
Rewrite the ratio using S: (S + 2ab) / (S − 2ab) = 25/4. Substitute ab = 21 to get (S + 42)/(S − 42) = 25/4. Cross-multiply to solve for S. Then compute S + 3ab = S + 63.

Step-by-Step Solution:
1) Let S = a^2 + b^2. 2) Use identities: (a + b)^2 = S + 2ab (a − b)^2 = S − 2ab 3) Given ratio: (S + 2ab) / (S − 2ab) = 25/4 4) Substitute ab = 21: (S + 42) / (S − 42) = 25/4 5) Cross-multiply: 4(S + 42) = 25(S − 42) 6) Expand: 4S + 168 = 25S − 1050 7) Rearrange: 168 + 1050 = 25S − 4S 1218 = 21S 8) Solve for S: S = 1218/21 = 58 9) Compute the required value: a^2 + b^2 + 3ab = S + 3ab = 58 + 63 = 121

Verification / Alternative check:
From S = 58 and ab = 21, we can check consistency: (a + b)^2 = S + 2ab = 58 + 42 = 100. (a − b)^2 = S − 2ab = 58 − 42 = 16. Ratio = 100/16 = 25/4, which matches the given condition exactly. Therefore S is correct, and the final computed value 121 is reliable.

Why Other Options Are Wrong:
• 115, 125, 127, 129: typically come from cross-multiplication mistakes or using 2ab as 21 instead of 42. • Some errors occur when students use (a − b)^2 = S + 2ab (wrong sign).

Common Pitfalls:
• Mixing up +2ab and −2ab in the identities. • Forgetting to multiply ab by 2 when substituting into S ± 2ab. • Arithmetic slips when moving terms across the equation (especially signs on 1050).

Final Answer:
121