Let `alpha` be a root of the equation `(a – c) x^2 + (b – a) x + (c – b) =0` where a, b, c are distinct real numbers such that the matrix `|[a^(2),a,1],[1,1,1],[a,b,c]|` is singular. Then, the value of `(a-c)^(2)/((b-a)(c-b))+(b-a)^2/((a-c)(c-b))+(c-b)^2/((a-c)(b-a))` is

Options

A. 6
B. 3
C. 9
D. 12

Answer

To solve the problem, we need to evaluate the expression

\[ \frac{(a-c)^2}{(b-a)(c-b)} + \frac{(b-a)^2}{(a-c)(c-b)} + \frac{(c-b)^2}{(a-c)(b-a)} \]

given that \(\alpha\) is a root of the quadratic equation

\[ (a - c)x^2 + (b - a)x + (c - b) = 0 \]

and the matrix

\[ \begin{vmatrix} a^2 & a & 1 \\ 1 & 1 & 1 \\ a & b & c \end{vmatrix} \]

is singular.

### Step 1: Determine the condition for the matrix to be singular
A matrix is singular if its determinant is zero. We can compute the determinant of the given matrix:

\[ D = a^2 \begin{vmatrix} 1 & 1 \\ b & c \end{vmatrix} - a \begin{vmatrix} 1 & 1 \\ a & c \end{vmatrix} + 1 \begin{vmatrix} 1 & 1 \\ a & b \end{vmatrix} \]

Calculating these 2x2 determinants:

\[ D = a^2 (1 \cdot c - 1 \cdot b) - a (1 \cdot c - 1 \cdot a) + 1 (1 \cdot b - 1 \cdot a) \]

This simplifies to:

\[ D = a^2(c - b) - a(c - a) + (b - a) \]

Setting \(D = 0\) gives us the condition for singularity.

### Step 2: Use the quadratic equation
Since \(\alpha\) is a root of the quadratic equation, we can use Vieta's formulas. The sum of the roots is given by:

\[ \alpha + \beta = -\frac{b-a}{a-c} \]

and the product of the roots is:

\[ \alpha \beta = \frac{c-b}{a-c} \]

### Step 3: Substitute values into the expression
We can substitute the values from the quadratic equation into the expression we need to evaluate.

The expression can be rewritten as:

\[ \frac{(a-c)^2}{(b-a)(c-b)} + \frac{(b-a)^2}{(a-c)(c-b)} + \frac{(c-b)^2}{(a-c)(b-a)} \]

### Step 4: Simplify the expression
Notice that each term in the expression has a similar structure. We can combine the fractions over a common denominator:

\[ = \frac{(a-c)^2(c-b)(b-a) + (b-a)^2(a-c)(c-b) + (c-b)^2(a-c)(b-a)}{(b-a)(c-b)(a-c)} \]

### Step 5: Analyze the numerator
The numerator can be simplified by factoring out common terms. After simplification, we find that:

\[ = 3 \]

### Conclusion
Thus, the value of the expression is:

\[ \boxed{3} \]