If one of the roots of `3x^(2)+11x+k=0` be reciprocal of the other, find value of `k`.

Options

A. 4
B. 3
C. 2
D. can't be determined

Answer

To find the value of \( k \) in the equation \( 3x^2 + 11x + k = 0 \) given that one root is the reciprocal of the other, we can follow these steps:

### Step 1: Understand the relationship between the roots
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). According to the problem, one root is the reciprocal of the other, which means:
\[ \beta = \frac{1}{\alpha} \]

### Step 2: Use the product of the roots
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the product of the roots \( \alpha \) and \( \beta \) can be expressed as:
\[ \alpha \cdot \beta = \frac{c}{a} \]
In our case, \( a = 3 \), \( b = 11 \), and \( c = k \). Therefore, we have:
\[ \alpha \cdot \beta = \frac{k}{3} \]

### Step 3: Substitute the reciprocal relationship into the product
Substituting \( \beta = \frac{1}{\alpha} \) into the product of the roots gives:
\[ \alpha \cdot \frac{1}{\alpha} = 1 \]
Thus, we can set up the equation:
\[ 1 = \frac{k}{3} \]

### Step 4: Solve for \( k \)
To find \( k \), multiply both sides of the equation by 3:
\[ k = 3 \cdot 1 = 3 \]

### Conclusion
The value of \( k \) is \( 3 \).

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