If (4a + 7b)(4c − 7d) = (4a − 7b)(4c + 7d), which of the following relations between a, b, c and d is correct (assuming all are nonzero)?

Introduction / Context:
This is an algebraic identity question connected with factorisation and symmetry. Given an equation involving products of linear expressions in a, b, c and d, you must deduce a proportional relation between these variables. Such problems train you to expand and compare coefficients, then interpret the resulting condition as a ratio relation.

Given Data / Assumptions:

• (4a + 7b)(4c − 7d) = (4a − 7b)(4c + 7d).
• a, b, c and d are nonzero real numbers.
• We want to know which ratio relation is implied by this equality.

Concept / Approach:
We can either expand both sides and compare terms, or we can rearrange the equality to factor out common expressions. Expansion is straightforward and quickly leads to a single simple condition involving ad and bc, which can then be rewritten as a/b = c/d.

Step-by-Step Solution:
1. Expand the left-hand side: (4a + 7b)(4c − 7d). 2. Left-hand side = 4a·4c + 4a·(−7d) + 7b·4c + 7b·(−7d). 3. This simplifies to 16ac − 28ad + 28bc − 49bd. 4. Expand the right-hand side: (4a − 7b)(4c + 7d). 5. Right-hand side = 4a·4c + 4a·7d + (−7b)·4c + (−7b)·7d. 6. This simplifies to 16ac + 28ad − 28bc − 49bd. 7. Set left-hand side equal to right-hand side: 16ac − 28ad + 28bc − 49bd = 16ac + 28ad − 28bc − 49bd. 8. Subtract 16ac and −49bd from both sides; they cancel. We get −28ad + 28bc = 28ad − 28bc. 9. Bring like terms to one side: −28ad − 28ad + 28bc + 28bc = 0, so −56ad + 56bc = 0. 10. Divide by 56 (nonzero): −ad + bc = 0, or bc = ad. 11. Since a, b, c, d are nonzero, divide both sides by bd: c/d = a/b. 12. Hence, a/b = c/d.

Verification / Alternative check:
You can pick simple values satisfying a/b = c/d, such as a = 2, b = 1, c = 4, d = 2. Substituting these into both sides of the original equation shows that the equality holds, which supports the derived relation.

Why Other Options Are Wrong:

• a/d = c/b or a/b = d/c are different ratio relations and do not follow from bc = ad in general.
• None of these is incorrect because we have clearly derived a/b = c/d from the given equation.

Common Pitfalls:
Some students attempt to cancel terms prematurely or incorrectly, or they mis-expand one of the products, leading to sign errors. It is important to expand carefully and systematically, then simplify step by step. Also, one must avoid dividing by zero; that is why we assume a, b, c and d are nonzero when forming ratios.

Final Answer:
The correct relation implied by the equation is a/b = c/d.