Suppose two quadratic equations `a_(1)x^(2)+b_(1)x+c_(1)=0` and `a_(2)x^(2)+b_(2)x+c_(2)=0` have a common root `alpha`, then
`a_(1)alpha^(2)+b_(1)alpha+c_(1)=0 " "` …..(1)
and `a_(2)alpha^(2)+b_(2)alpha+c_(2)=0 " "` ……(2)
Eliminating `alpha` using crose - multiplication method gives us the condition for a common root Solving two equations simultaneously, the common root can be obtained. Now, consider three quadratic equations, `x^(2)-2r p_(r )x+r=0, r=1, 2, 3` given that each pair has exactly one root common.
In the notations of above problem, `(gamma)/(alpha)` is equal to

Options

A. 2
B. 3
C. `1//2`
D. `1//3`

Answer

To solve the problem step by step, we will first analyze the given quadratic equations and then derive the required relationship between the roots.

### Step 1: Write the Quadratic Equations
We have three quadratic equations given as:
1. \( x^2 - 2p_1x + 1 = 0 \) (for \( r = 1 \))
2. \( x^2 - 4p_2x + 2 = 0 \) (for \( r = 2 \))
3. \( x^2 - 6p_3x + 3 = 0 \) (for \( r = 3 \))

### Step 2: Identify Common Roots
We know that each pair of equations has exactly one root in common. Let’s denote the common roots as follows:
- Let \( \beta \) be the common root of the first two equations.
- Let \( \gamma \) be the common root of the second and third equations.
- Let \( \alpha \) be the common root of the first and third equations.

### Step 3: Use Vieta's Formulas
From Vieta's formulas, we can express the relationships between the roots and coefficients of the quadratic equations:
- For the first equation: \( \beta + \alpha = 2p_1 \) and \( \beta \alpha = 1 \)
- For the second equation: \( \beta + \gamma = 4p_2 \) and \( \beta \gamma = 2 \)
- For the third equation: \( \gamma + \alpha = 6p_3 \) and \( \gamma \alpha = 3 \)

### Step 4: Express Roots in Terms of Each Other
From the product of the roots:
1. From \( \beta \alpha = 1 \), we can express \( \alpha = \frac{1}{\beta} \).
2. From \( \beta \gamma = 2 \), we can express \( \gamma = \frac{2}{\beta} \).
3. From \( \gamma \alpha = 3 \), substituting \( \alpha \) gives \( \gamma \cdot \frac{1}{\beta} = 3 \) or \( \gamma = 3\beta \).

### Step 5: Set Up Equations
Now we have:
- \( \gamma = \frac{2}{\beta} \) and \( \gamma = 3\beta \).

Equating these two expressions for \( \gamma \):
\[ \frac{2}{\beta} = 3\beta \]

### Step 6: Solve for \( \beta \)
Multiply both sides by \( \beta \):
\[ 2 = 3\beta^2 \]
\[ \beta^2 = \frac{2}{3} \]
\[ \beta = \pm \sqrt{\frac{2}{3}} = \pm \frac{\sqrt{6}}{3} \]

### Step 7: Find \( \alpha \) and \( \gamma \)
Using \( \beta \):
1. \( \alpha = \frac{1}{\beta} = \pm \frac{3}{\sqrt{6}} = \pm \frac{\sqrt{6}}{2} \).
2. \( \gamma = 3\beta = \pm 3 \cdot \frac{\sqrt{6}}{3} = \pm \sqrt{6} \).

### Step 8: Find \( \frac{\gamma}{\alpha} \)
Now we can find the ratio:
\[ \frac{\gamma}{\alpha} = \frac{\pm \sqrt{6}}{\pm \frac{\sqrt{6}}{2}} = 2 \]

### Conclusion
Thus, the value of \( \frac{\gamma}{\alpha} \) is \( 2 \).