What is the value of k, if one zero of the polynomial (k- 1)`x^2` - 10x + 3 is reciprocal of the other?

Options

A. 4
B. 5
C. -1
D. 0

Answer

To find the value of \( k \) such that one zero of the polynomial \( (k - 1)x^2 - 10x + 3 \) is the reciprocal of the other, we can follow these steps:

### Step 1: Understand the Polynomial
The given polynomial can be expressed as:
\[ (k - 1)x^2 - 10x + 3 = 0 \]
This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = k - 1 \)
- \( b = -10 \)
- \( c = 3 \)

### Step 2: Define the Roots
Let the roots of the polynomial be \( \alpha \) and \( \beta \). According to the problem, one root is the reciprocal of the other, which means:
\[ \beta = \frac{1}{\alpha} \]

### Step 3: Use the Relationship of Roots
For a quadratic equation, the product of the roots \( \alpha \) and \( \beta \) is given by:
\[ \alpha \cdot \beta = \frac{c}{a} \]
Substituting the values of \( c \) and \( a \):
\[ \alpha \cdot \frac{1}{\alpha} = \frac{3}{k - 1} \]
This simplifies to:
\[ 1 = \frac{3}{k - 1} \]

### Step 4: Solve for \( k \)
Cross-multiplying gives:
\[ k - 1 = 3 \]
Adding 1 to both sides:
\[ k = 4 \]

### Conclusion
The value of \( k \) is:
\[ \boxed{4} \]

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