If \(a^{2}+b^{2}=5 a b\), then the value of \(\left(\frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}\right)\) is : (a) 32 (b) 16 (c) 23 (d) -23

Calculation of the Given Expression

Given the equation \(a^{2}+b^{2}=5ab\), we are tasked with determining the value of the expression \(\left(\frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}\right)\).

Step 1: Simplify the Given Relation

Starting with the given equation, we divide both sides by \(ab\) to simplify:

\[
\frac{a^2 + b^2}{ab} = 5
\]

This leads to:

\[
\frac{a}{b} + \frac{b}{a} = 5
\]

Step 2: Square Both Sides

To find the value of \(\left(\frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}\right)\), we square both sides of the simplified equation:

\[
\left(\frac{a}{b} + \frac{b}{a}\right)^2 = 5^2
\]

This yields:

\[
\frac{a^2}{b^2} + 2\left(\frac{a}{b}\cdot\frac{b}{a}\right) + \frac{b^2}{a^2} = 25
\]

Given that \(\frac{a}{b}\cdot\frac{b}{a} = 1\), we simplify further:

\[
\frac{a^2}{b^2} + \frac{b^2}{a^2} + 2 = 25
\]

Step 3: Isolate the Target Expression

Subtracting 2 from both sides to isolate the expression gives us:

\[
\frac{a^2}{b^2} + \frac{b^2}{a^2} = 25 – 2 = 23
\]

Conclusion

Therefore, the value of the expression \(\left(\frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}\right)\) is \(\textbf{23}\), making the correct answer:

(c) \(\textbf{23}\)

This solution methodically derives the value of the given expression by leveraging the initial condition and algebraic manipulation, leading to a clear and logical conclusion.