If `a ^(2) + 1 =9a, (a ne 0)` then the value of `(a) ^(2) + (1)/((a) ^(2))`

Options

A. 81
B. 18
C. 79
D. 83

Answer

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

1. **Given Equation**:
\[ a^2 + 1 = 9a \]

2. **Rearranging the Equation**:
We can rearrange this equation to isolate \(a^2\):
\[ a^2 - 9a + 1 = 0 \]

3. **Finding \( \frac{1}{a^2} \)**:
We need to find the value of \( a^2 + \frac{1}{a^2} \). To do this, we can first find \( a + \frac{1}{a} \).

4. **Dividing the Original Equation**:
From the original equation, we can express \(a + \frac{1}{a}\):
\[ a + \frac{1}{a} = 9 \]

5. **Squaring Both Sides**:
Now, we square both sides to find \( a^2 + \frac{1}{a^2} \):
\[ \left(a + \frac{1}{a}\right)^2 = 9^2 \]
This expands to:
\[ a^2 + 2 + \frac{1}{a^2} = 81 \]

6. **Isolating \( a^2 + \frac{1}{a^2} \)**:
Now, we can isolate \( a^2 + \frac{1}{a^2} \):
\[ a^2 + \frac{1}{a^2} = 81 - 2 \]
Simplifying gives:
\[ a^2 + \frac{1}{a^2} = 79 \]

Thus, the final value of \( a^2 + \frac{1}{a^2} \) is:
\[ \boxed{79} \]