If one of the roots of the equation a(b -c) `x^(2) + b (c -a) x + c (a - b) = 0` is 1. then what is the second root?

Options

A. `(b(c-a))/(a(b-c))`
B. `(b(c-a))/(a(b-c))`
C. `(c(a-b))/(a(b-c))`
D. `-(c(a-b))/(a(b-c))`

Answer

To find the second root of the quadratic equation given that one root is 1, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given quadratic equation**:
The equation is given as:
\[ a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \]

2. **Assume the roots**:
Let the roots of the equation be \( \alpha \) and \( \beta \). We know that one of the roots, \( \alpha \), is given as 1.

3. **Use the relationship between the roots**:
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the product of the roots \( \alpha \) and \( \beta \) can be expressed as:
\[ \alpha \cdot \beta = \frac{c}{a} \]
Here, \( c \) is the constant term and \( a \) is the coefficient of \( x^2 \).

4. **Substituting the known root**:
Since \( \alpha = 1 \), we can substitute this into the product of the roots:
\[ 1 \cdot \beta = \frac{c}{a} \]
This simplifies to:
\[ \beta = \frac{c}{a} \]

5. **Identify \( c \) and \( a \)**:
From the given equation:
- \( a = a(b - c) \)
- \( c = c(a - b) \)

6. **Substituting values**:
Now substituting the expressions for \( c \) and \( a \):
\[ \beta = \frac{c(a - b)}{a(b - c)} \]

7. **Final expression for the second root**:
Thus, the second root \( \beta \) is:
\[ \beta = \frac{c(a - b)}{a(b - c)} \]

### Conclusion:
The second root of the equation is:
\[ \beta = \frac{c(a - b)}{a(b - c)} \]