If `a in Z` and the equation `(x-a)(x-10)+1=0` has integral roots, then values of `a` are

Options

A. 10, 8
B. 12, 10
C. 12, 8
D. none of these

Explanation

To solve the problem, we need to determine the values of \( a \) such that the equation

\[ (x - a)(x - 10) + 1 = 0 \]

has integral roots. Let's break this down step by step.

### Step 1: Rearranging the Equation
First, we can rearrange the equation to isolate the product:

\[ (x - a)(x - 10) = -1 \]

### Step 2: Analyzing the Product
The left side of the equation is a product of two expressions, \( (x - a) \) and \( (x - 10) \). Since the right side equals \(-1\), we need to consider the pairs of integers that multiply to \(-1\). The pairs are:

1. \( (1, -1) \)
2. \( (-1, 1) \)

### Step 3: Solving the First Pair
Let's take the first pair \( (1, -1) \):

1. Set \( x - a = 1 \) and \( x - 10 = -1 \).

From \( x - 10 = -1 \):

\[ x = 9 \]

Now substituting \( x = 9 \) into \( x - a = 1 \):

\[ 9 - a = 1 \implies a = 8 \]

### Step 4: Solving the Second Pair
Now, let's take the second pair \( (-1, 1) \):

1. Set \( x - a = -1 \) and \( x - 10 = 1 \).

From \( x - 10 = 1 \):

\[ x = 11 \]

Now substituting \( x = 11 \) into \( x - a = -1 \):

\[ 11 - a = -1 \implies a = 12 \]

### Step 5: Conclusion
The possible integral values of \( a \) that satisfy the equation are:

\[ a = 8 \quad \text{and} \quad a = 12 \]

### Final Answer
Thus, the values of \( a \) are \( 8 \) and \( 12 \).

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