the equation of the radical axis of the two circles `7x^2+7y^2-7x+14y+18=0` and `4x^2+4y^2-7x+8y+20=0` is given by

Options

A. `3x ^(2) + 3y ^(2) - 6y -2=0`
B. `21x-68=0`
C. `x - 2y -5 =0`
D. none of these

Answer

To find the equation of the radical axis of the two circles given by the equations:

1. \( 7x^2 + 7y^2 - 7x + 14y + 18 = 0 \)
2. \( 4x^2 + 4y^2 - 7x + 8y + 20 = 0 \)

we will follow these steps:

### Step 1: Simplify the equations of the circles

For the first circle:

\[ 7x^2 + 7y^2 - 7x + 14y + 18 = 0 \]

Dividing the entire equation by 7:

\[ x^2 + y^2 - x + 2y + \frac{18}{7} = 0 \quad \text{(Let this be } s_1\text{)} \]

For the second circle:

\[ 4x^2 + 4y^2 - 7x + 8y + 20 = 0 \]

Dividing the entire equation by 4:

\[ x^2 + y^2 - \frac{7}{4}x + 2y + 5 = 0 \quad \text{(Let this be } s_2\text{)} \]

### Step 2: Write the equations for \(s_1\) and \(s_2\)

Now we have:

- \( s_1: x^2 + y^2 - x + 2y + \frac{18}{7} = 0 \)
- \( s_2: x^2 + y^2 - \frac{7}{4}x + 2y + 5 = 0 \)

### Step 3: Find the equation of the radical axis

The equation of the radical axis is given by \( s_1 - s_2 = 0 \).

Subtracting \(s_2\) from \(s_1\):

\[ (x^2 + y^2 - x + 2y + \frac{18}{7}) - (x^2 + y^2 - \frac{7}{4}x + 2y + 5) = 0 \]

This simplifies to:

\[ -x + \frac{7}{4}x + \frac{18}{7} - 5 = 0 \]

### Step 4: Combine like terms

Combining the \(x\) terms:

\[ \left(-1 + \frac{7}{4}\right)x + \left(\frac{18}{7} - 5\right) = 0 \]

Calculating \(-1 + \frac{7}{4}\):

\[ -1 = -\frac{4}{4} \implies -1 + \frac{7}{4} = \frac{3}{4} \]

Now for the constant terms:

\[ \frac{18}{7} - 5 = \frac{18}{7} - \frac{35}{7} = -\frac{17}{7} \]

So we have:

\[ \frac{3}{4}x - \frac{17}{7} = 0 \]

### Step 5: Clear the fractions

To eliminate the fractions, multiply through by 28 (the least common multiple of 4 and 7):

\[ 28 \left(\frac{3}{4}x\right) - 28 \left(\frac{17}{7}\right) = 0 \]

This gives:

\[ 21x - 68 = 0 \]

### Final Step: Write the equation of the radical axis

Thus, the equation of the radical axis is:

\[ 21x - 68 = 0 \]