If `a ^(2) + b ^(2) = 5 ab,` then the value of `((a ^(2))/( b ^(2)) + (b ^(2))/( a ^(2)))` is :

Options

A. 32
B. 16
C. 23
D. `-23`

Answer

To solve the problem, we start with the given equation:

**Step 1:** We are given that \( a^2 + b^2 = 5ab \).

**Step 2:** We need to find the value of \( \frac{a^2}{b^2} + \frac{b^2}{a^2} \). We can rewrite this expression using a common denominator:

\[ \frac{a^2}{b^2} + \frac{b^2}{a^2} = \frac{a^4 + b^4}{a^2b^2} \]

**Step 3:** To find \( a^4 + b^4 \), we can use the identity:

\[ a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 \]

**Step 4:** We know from Step 1 that \( a^2 + b^2 = 5ab \). Therefore, we can square both sides:

\[ (a^2 + b^2)^2 = (5ab)^2 = 25a^2b^2 \]

**Step 5:** Now substituting this back into the identity we derived in Step 3:

\[ a^4 + b^4 = 25a^2b^2 - 2a^2b^2 = 23a^2b^2 \]

**Step 6:** Now we substitute \( a^4 + b^4 \) back into our expression from Step 2:

\[ \frac{a^4 + b^4}{a^2b^2} = \frac{23a^2b^2}{a^2b^2} \]

**Step 7:** Simplifying this gives us:

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

Thus, the value of \( \frac{a^2}{b^2} + \frac{b^2}{a^2} \) is **23**.