If one root of `5x^(2)+13x+k=0` be the reciprocal of the other root then the value of k is

Options

A. 0
B. 1
C. 2
D. 5

Explanation

To find the value of \( k \) in the quadratic equation \( 5x^2 + 13x + k = 0 \) given that one root is the reciprocal of the other root, we can follow these steps:

### Step 1: Understand the properties of the roots
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). According to the problem, if one root is \( \alpha \), then the other root \( \beta \) is \( \frac{1}{\alpha} \).

### Step 2: Use the relationship between the roots and coefficients
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum and product of the roots can be expressed as:
- Sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- Product of the roots \( \alpha \cdot \beta = \frac{c}{a} \)

Here, \( a = 5 \), \( b = 13 \), and \( c = k \).

### Step 3: Calculate the sum of the roots
Using the sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{13}{5} \]
Substituting \( \beta = \frac{1}{\alpha} \):
\[ \alpha + \frac{1}{\alpha} = -\frac{13}{5} \]

### Step 4: Calculate the product of the roots
Using the product of the roots:
\[ \alpha \cdot \beta = \frac{c}{a} = \frac{k}{5} \]
Substituting \( \beta = \frac{1}{\alpha} \):
\[ \alpha \cdot \frac{1}{\alpha} = 1 = \frac{k}{5} \]

### Step 5: Solve for \( k \)
From the product of the roots, we have:
\[ 1 = \frac{k}{5} \]
Multiplying both sides by 5 gives:
\[ k = 5 \]

### Final Answer
Thus, the value of \( k \) is \( 5 \).

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