If `P=7+4sqrt(3)` and `PQ=1`, then what is the value of `(1)/(P^(2))+(1)/(Q^(2))` ?

Options

A. `196`
B. `194`
C. `206`
D. `182`

Answer

To solve the problem, we need to find the value of \( \frac{1}{P^2} + \frac{1}{Q^2} \) given that \( P = 7 + 4\sqrt{3} \) and \( PQ = 1 \).

### Step-by-step Solution:

1. **Find Q in terms of P**:
Since \( PQ = 1 \), we can express \( Q \) as:
\[ Q = \frac{1}{P} = \frac{1}{7 + 4\sqrt{3}} \]

2. **Rationalize the denominator of Q**:
To simplify \( Q \), we multiply the numerator and denominator by the conjugate of the denominator:
\[ Q = \frac{1 \cdot (7 - 4\sqrt{3})}{(7 + 4\sqrt{3})(7 - 4\sqrt{3})} \]
The denominator simplifies as follows:
\[ (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 = 49 - 48 = 1 \]
Thus, we have:
\[ Q = 7 - 4\sqrt{3} \]

3. **Calculate \( P^2 \) and \( Q^2 \)**:
Now, we need to find \( P^2 \) and \( Q^2 \):
\[ P^2 = (7 + 4\sqrt{3})^2 = 7^2 + 2 \cdot 7 \cdot 4\sqrt{3} + (4\sqrt{3})^2 = 49 + 56\sqrt{3} + 48 = 97 + 56\sqrt{3} \]
\[ Q^2 = (7 - 4\sqrt{3})^2 = 7^2 - 2 \cdot 7 \cdot 4\sqrt{3} + (4\sqrt{3})^2 = 49 - 56\sqrt{3} + 48 = 97 - 56\sqrt{3} \]

4. **Calculate \( \frac{1}{P^2} + \frac{1}{Q^2} \)**:
Using the formula \( \frac{1}{P^2} + \frac{1}{Q^2} = \frac{Q^2 + P^2}{P^2 Q^2} \):
\[ P^2 + Q^2 = (97 + 56\sqrt{3}) + (97 - 56\sqrt{3}) = 194 \]
Next, we calculate \( P^2 Q^2 \):
\[ P^2 Q^2 = (P \cdot Q)^2 = (PQ)^2 = 1^2 = 1 \]
Therefore:
\[ \frac{1}{P^2} + \frac{1}{Q^2} = \frac{P^2 + Q^2}{P^2 Q^2} = \frac{194}{1} = 194 \]

### Final Answer:
The value of \( \frac{1}{P^2} + \frac{1}{Q^2} \) is \( 194 \).