If `a^2+b^2=45` and `ab=18`, then the value of `(1)/(a)+(1)/(b)` is

Options

A. `(5)/(7)`
B. `(7)/(8)`
C. `+-1/2`
D. `+-1/3`

Answer

To solve the problem, we need to find the value of \( \frac{1}{a} + \frac{1}{b} \) given that \( a^2 + b^2 = 45 \) and \( ab = 18 \).

### Step 1: Use the identity for \( a^2 + b^2 \)
We know the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting the known values into this identity:
\[ 45 = (a + b)^2 - 2 \cdot 18 \]

### Step 2: Simplify the equation
Now, simplify the equation:
\[ 45 = (a + b)^2 - 36 \]
Adding 36 to both sides gives:
\[ 45 + 36 = (a + b)^2 \]
\[ 81 = (a + b)^2 \]

### Step 3: Take the square root
Taking the square root of both sides, we find:
\[ a + b = 9 \quad \text{or} \quad a + b = -9 \]
Since we are looking for positive values of \( a \) and \( b \), we will take \( a + b = 9 \).

### Step 4: Find \( \frac{1}{a} + \frac{1}{b} \)
We can express \( \frac{1}{a} + \frac{1}{b} \) in terms of \( a + b \) and \( ab \):
\[ \frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab} \]
Substituting the values we have:
\[ \frac{1}{a} + \frac{1}{b} = \frac{9}{18} \]

### Step 5: Simplify the fraction
Now, simplify the fraction:
\[ \frac{9}{18} = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \frac{1}{a} + \frac{1}{b} \) is \( \frac{1}{2} \).

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