If `p + q = 7 and pq = 12`, then will is the value of `1/(p^2) + 1/(q^2)` ?

Options

A. `1//6`
B. `25//144`
C. `49//144`
D. `7//12`

Answer

To solve the problem, we need to find the value of \( \frac{1}{p^2} + \frac{1}{q^2} \) given that \( p + q = 7 \) and \( pq = 12 \).

### Step-by-Step Solution:

1. **Start with the given equations:**
\[ p + q = 7 \quad \text{(1)} \]
\[ pq = 12 \quad \text{(2)} \]

2. **Rewrite the expression \( \frac{1}{p^2} + \frac{1}{q^2} \):**
\[ \frac{1}{p^2} + \frac{1}{q^2} = \frac{q^2 + p^2}{p^2 q^2} \]

3. **Use the identity for \( p^2 + q^2 \):**
We know that:
\[ p^2 + q^2 = (p + q)^2 - 2pq \]
Substitute the values from equations (1) and (2):
\[ p^2 + q^2 = (7)^2 - 2(12) \]
\[ = 49 - 24 = 25 \]

4. **Calculate \( p^2 q^2 \):**
Since \( pq = 12 \), we have:
\[ p^2 q^2 = (pq)^2 = 12^2 = 144 \]

5. **Substitute back into the expression:**
Now substitute \( p^2 + q^2 \) and \( p^2 q^2 \) into the expression:
\[ \frac{1}{p^2} + \frac{1}{q^2} = \frac{p^2 + q^2}{p^2 q^2} = \frac{25}{144} \]

6. **Final answer:**
Thus, the value of \( \frac{1}{p^2} + \frac{1}{q^2} \) is:
\[ \frac{25}{144} \]