To solve the problem, let's assume the given condition is (\frac{a+b}{a-b} = \frac{5}{4}) (a typical setup for this type of problem). Here's the step-by-step solution:

Step 1: Cross-multiply to simplify the given ratio

(\frac{a+b}{a-b} = \frac{5}{4})
Cross-multiplying gives:
(4(a + b) = 5(a - b))

Step 2: Expand and rearrange terms

(4a + 4b = 5a - 5b)
Bring like terms together:
(5b + 4b = 5a - 4a)
(9b = a) → (a = 9b)

Step 3: Compute (\frac{a^2 + b^2}{a^2 - b^2})

Substitute (a = 9b):
(\frac{(9b)^2 + b^2}{(9b)^2 - b^2} = \frac{81b^2 + b^2}{81b^2 - b^2} = \frac{82b^2}{80b^2})

Cancel (b^2) (since (b \neq 0)):
(\frac{82}{80} = \frac{41}{40})

Answer: (\boxed{\dfrac{41}{40}})

作者声明:本文包含人工智能生成内容。