Find the value of k for which the roots of the equation `3x^2-10x+k=0` are reciprocal of each other

Answer

To find the value of \( k \) for which the roots of the equation \( 3x^2 - 10x + k = 0 \) are reciprocal of each other, we can follow these steps:

### Step 1: Understand the condition for reciprocal roots
For the roots \( \alpha \) and \( \beta \) of a quadratic equation to be reciprocal, the product of the roots must equal 1. That is, \( \alpha \cdot \beta = 1 \).

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a quadratic equation of the form \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \cdot \beta = \frac{c}{a} \)

In our case, \( a = 3 \), \( b = -10 \), and \( c = k \).

### Step 3: Set up the equation for the product of roots
From Vieta's formulas, we have:
\[ \alpha \cdot \beta = \frac{k}{3} \]
Since we want the product of the roots to be 1 (because they are reciprocal), we set up the equation:
\[ \frac{k}{3} = 1 \]

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

### Conclusion
Thus, the value of \( k \) for which the roots of the equation \( 3x^2 - 10x + k = 0 \) are reciprocal of each other is:
\[ \boxed{3} \]