Assertion (A): The roots of the equation `a(b-c)x^(2)+b(c-a)x+c(a-b)=0` are 1, `(c(a-b))/(a(b-c))`
Reason (R): If a+b+c=0 then the roots of `ax^(2)+bx+c=0` are 1, `(c)/(a)`

Options

A. Both A, R are true and R explain Assertion
B. Both A, R are true but R does't explain A
C. A is true R is false
D. A is false R is true

Answer

To solve the given problem, we need to analyze both the assertion (A) and the reason (R) provided in the question.

### Step 1: Analyze the Reason (R)
The reason states that if \( a + b + c = 0 \), then the roots of the equation \( ax^2 + bx + c = 0 \) are \( 1 \) and \( \frac{c}{a} \).

1. **Substituting \( x = 1 \)**:
- We substitute \( x = 1 \) into the equation \( ax^2 + bx + c \):
\[ a(1)^2 + b(1) + c = a + b + c \]
- Since \( a + b + c = 0 \), we conclude that \( x = 1 \) is indeed a root.

2. **Using the Product of Roots**:
- The product of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \frac{c}{a} \).
- If one root is \( 1 \), we can express the product of the roots as:
\[ 1 \cdot r = \frac{c}{a} \]
where \( r \) is the other root. Thus, \( r = \frac{c}{a} \).

### Conclusion for Reason (R):
Since both parts of the reason are verified, we conclude that the reason (R) is true.

### Step 2: Analyze the Assertion (A)
The assertion states that the roots of the equation \( a(b-c)x^2 + b(c-a)x + c(a-b) = 0 \) are \( 1 \) and \( \frac{c(a-b)}{a(b-c)} \).

1. **Sum of Coefficients**:
- We find the sum of the coefficients of the equation:
\[ a(b-c) + b(c-a) + c(a-b) \]
- Simplifying this:
\[ ab - ac + bc - ab + ca - bc = 0 \]
- Since the sum of the coefficients is \( 0 \), \( x = 1 \) is a root of the equation.

2. **Product of Roots**:
- The product of the roots for the quadratic equation \( a(b-c)x^2 + b(c-a)x + c(a-b) = 0 \) is given by:
\[ \frac{c(a-b)}{a(b-c)} \]
- Since we already established that one root is \( 1 \), we can express the product of the roots as:
\[ 1 \cdot r = \frac{c(a-b)}{a(b-c)} \]
- Therefore, the second root \( r \) is indeed \( \frac{c(a-b)}{a(b-c)} \).

### Conclusion for Assertion (A):
Since both parts of the assertion are verified, we conclude that the assertion (A) is true.

### Final Conclusion:
Both the assertion (A) and reason (R) are true, and the reason explains the assertion. Therefore, the final answer is that both statements are true.

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