If the roots of `a_1x^2 + b_1x+ c_1 = 0` are `alpha_1 ,beta_ 1` and those of `a_2x^2+b_2x+c_2=0` are `alpha_2,beta_2` such that `alpha_1alpha_2=beta_1beta_2=1` then

Options

A. `(a_(1))/(a_(2))=(b_(1))/(b_(2))=(c_(1))/(c_(2))`
B. `(a_(1))/(c_(2))=(b_(1))/(b_(2))=(c_(1))/(a_(2))`
C. `a_(1) a_(2) = b_(1) b_(2) = c_(1) c_(2)`
D. none of these

Answer

To solve the problem, we will analyze the given quadratic equations and their roots step by step.

### Step 1: Identify the roots and their relationships
We have two quadratic equations:
1. \( a_1x^2 + b_1x + c_1 = 0 \) with roots \( \alpha_1 \) and \( \beta_1 \)
2. \( a_2x^2 + b_2x + c_2 = 0 \) with roots \( \alpha_2 \) and \( \beta_2 \)

From Vieta's formulas, we know:
- For the first equation:
- \( \alpha_1 + \beta_1 = -\frac{b_1}{a_1} \) (sum of roots)
- \( \alpha_1 \beta_1 = \frac{c_1}{a_1} \) (product of roots)

- For the second equation:
- \( \alpha_2 + \beta_2 = -\frac{b_2}{a_2} \) (sum of roots)
- \( \alpha_2 \beta_2 = \frac{c_2}{a_2} \) (product of roots)

### Step 2: Use the given conditions
We are given:
- \( \alpha_1 \alpha_2 = 1 \)
- \( \beta_1 \beta_2 = 1 \)

### Step 3: Multiply the products of the roots
Now, we can multiply the products of the roots:
\[ \alpha_1 \beta_1 \cdot \alpha_2 \beta_2 = \left(\frac{c_1}{a_1}\right) \cdot \left(\frac{c_2}{a_2}\right) \]
Substituting the values we have:
\[ (\alpha_1 \alpha_2)(\beta_1 \beta_2) = 1 \cdot 1 = 1 \]
Thus,
\[ 1 = \frac{c_1 c_2}{a_1 a_2} \]
Rearranging gives us:
\[ c_1 c_2 = a_1 a_2 \]

### Step 4: Express relationships between coefficients
Now, we can express relationships between the coefficients:
From the previous step, we can also write:
\[ \frac{c_1}{a_1} = \frac{a_2}{c_2} \]
This gives us our first equation.

### Step 5: Find another relationship using sums of roots
Next, we can find another relationship using the sums of the roots:
\[ \alpha_1 \beta_1 (\alpha_2 + \beta_2) = \frac{c_1}{a_1} \left(-\frac{b_2}{a_2}\right) \]
Substituting \( \alpha_1 \beta_1 = \frac{c_1}{a_1} \):
\[ \frac{c_1}{a_1} (\alpha_2 + \beta_2) = \frac{c_1}{a_1} \left(-\frac{b_2}{a_2}\right) \]
Cancelling \( \frac{c_1}{a_1} \) (assuming \( c_1 \neq 0 \)):
\[ \alpha_2 + \beta_2 = -\frac{b_2}{a_2} \]

### Step 6: Final relationships
From the relationships derived, we can summarize:
1. \( \frac{c_1}{a_1} = \frac{a_2}{c_2} \)
2. \( \frac{c_1}{a_2} = \frac{b_1}{b_2} \)

Thus, we have established the relationships between the coefficients of the two quadratic equations.