For the circles `3x^2+3y^2+x+2y-1=0 , 2x^2+2y^2+2x-y-1=0` radical axis is

Options

A. 4x-7y=1
B. 3x+y-2=0
C. x-3y=0
D. 4x-7y=5

Answer

To find the radical axis of the given circles, we will follow these steps:

### Step 1: Write the equations of the circles
The equations of the circles are given as:
1. \( S_1: 3x^2 + 3y^2 + x + 2y - 1 = 0 \)
2. \( S_2: 2x^2 + 2y^2 + 2x - y - 1 = 0 \)

### Step 2: Simplify the equations
We can simplify both equations by dividing them by their respective coefficients.

For \( S_1 \):
\[ S_1: \frac{3x^2 + 3y^2 + x + 2y - 1}{3} = 0 \implies x^2 + y^2 + \frac{x}{3} + \frac{2y}{3} - \frac{1}{3} = 0 \]

For \( S_2 \):
\[ S_2: \frac{2x^2 + 2y^2 + 2x - y - 1}{2} = 0 \implies x^2 + y^2 + x - \frac{y}{2} - \frac{1}{2} = 0 \]

### Step 3: Set up the equation for the radical axis
The radical axis can be found using the condition \( S_1 - S_2 = 0 \).

Subtract \( S_2 \) from \( S_1 \):
\[ S_1 - S_2 = \left( x^2 + y^2 + \frac{x}{3} + \frac{2y}{3} - \frac{1}{3} \right) - \left( x^2 + y^2 + x - \frac{y}{2} - \frac{1}{2} \right) = 0 \]

### Step 4: Simplify the equation
Now, simplifying the equation:
\[ \frac{x}{3} - x + \frac{2y}{3} + \frac{y}{2} - \frac{1}{3} + \frac{1}{2} = 0 \]

Combine like terms:
- The \( x \) terms: \( \frac{x}{3} - x = \frac{x - 3x}{3} = -\frac{2x}{3} \)
- The \( y \) terms: \( \frac{2y}{3} + \frac{y}{2} \)
- The constants: \( -\frac{1}{3} + \frac{1}{2} = \frac{-2 + 3}{6} = \frac{1}{6} \)

So we have:
\[ -\frac{2x}{3} + \left(\frac{2y}{3} + \frac{3y}{6}\right) + \frac{1}{6} = 0 \]

Finding a common denominator for \( y \):
\[ -\frac{2x}{3} + \left(\frac{4y}{6} + \frac{3y}{6}\right) + \frac{1}{6} = 0 \]
\[ -\frac{2x}{3} + \frac{7y}{6} + \frac{1}{6} = 0 \]

### Step 5: Clear the fractions
Multiply through by 6 to eliminate the denominators:
\[ -4x + 7y + 1 = 0 \]

### Step 6: Rearranging the equation
Rearranging gives us the equation of the radical axis:
\[ 4x - 7y = -1 \quad \text{or} \quad 4x - 7y = 1 \]

### Final Answer
Thus, the equation of the radical axis is:
\[ 4x - 7y = 1 \]