The values of the parameter a for which the quadratic equations `(1-2a) x^(2) - 6ax - 1 =0 and ax^(2) - x + 1 = 0` have at least one root in common, are

Options

A. `0, (1)/(2)`
B. `(1)/(2), (2)/(9)`
C. `(2)/(9)`
D. `0, (1)/(2), (2)/(9)`

Explanation

To find the values of the parameter \( a \) for which the quadratic equations

\[ (1-2a)x^2 - 6ax - 1 = 0 \]

and

\[ ax^2 - x + 1 = 0 \]

have at least one root in common, we can follow these steps:

### Step 1: Identify the Roots
Let \( r \) be the common root of both equations. Then, we can express \( r \) in terms of \( a \) from both equations.

### Step 2: Substitute the Common Root
Substituting \( r \) into the first equation:

\[ (1-2a)r^2 - 6ar - 1 = 0 \tag{1} \]

Substituting \( r \) into the second equation:

\[ ar^2 - r + 1 = 0 \tag{2} \]

### Step 3: Solve for \( r^2 \)
From equation (2), we can express \( r^2 \) in terms of \( r \) and \( a \):

\[ ar^2 = r - 1 \implies r^2 = \frac{r - 1}{a} \tag{3} \]

### Step 4: Substitute \( r^2 \) in Equation (1)
Now substitute equation (3) into equation (1):

\[ (1-2a)\left(\frac{r - 1}{a}\right) - 6ar - 1 = 0 \]

### Step 5: Simplify the Equation
Multiply through by \( a \) to eliminate the fraction:

\[ (1-2a)(r - 1) - 6a^2r - a = 0 \]

Expanding this gives:

\[ (1 - 2a)r - (1 - 2a) - 6a^2r - a = 0 \]

Combine like terms:

\[ (1 - 2a - 6a^2)r - (1 - 2a + a) = 0 \]

### Step 6: Set the Coefficient of \( r \) to Zero
For the equations to have a common root, the coefficients must equal zero:

1. \( 1 - 2a - 6a^2 = 0 \)
2. \( -1 + a + 2a = 0 \)

### Step 7: Solve the First Equation
From \( 1 - 2a - 6a^2 = 0 \):

\[ 6a^2 + 2a - 1 = 0 \]

Using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ a = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 6 \cdot (-1)}}{2 \cdot 6} \]
\[ = \frac{-2 \pm \sqrt{4 + 24}}{12} \]
\[ = \frac{-2 \pm \sqrt{28}}{12} \]
\[ = \frac{-2 \pm 2\sqrt{7}}{12} \]
\[ = \frac{-1 \pm \sqrt{7}}{6} \]

### Step 8: Solve the Second Equation
From \( -1 + 3a = 0 \):

\[ 3a = 1 \implies a = \frac{1}{3} \]

### Step 9: Combine Solutions
The values of \( a \) for which the quadratic equations have at least one root in common are:

\[ a = \frac{-1 + \sqrt{7}}{6}, \quad a = \frac{-1 - \sqrt{7}}{6}, \quad a = \frac{1}{3} \]

### Final Answer
The values of \( a \) are \( \frac{-1 + \sqrt{7}}{6} \), \( \frac{-1 - \sqrt{7}}{6} \), and \( \frac{1}{3} \).

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