The equation of the radical axis of the pair of circles `7x^2+7y^2-7x+14y+18=0` and `4x^2+4y^2-7x+8y+20=0`

Options

A. x-2y-5=0
B. 21x-68=0
C. 2x-y+5=0
D. 23x-68=0

Answer

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

### Step 1: Write the equations of the circles
The equations of the circles are given as:
1. \( S_1: 7x^2 + 7y^2 - 7x + 14y + 18 = 0 \)
2. \( S_2: 4x^2 + 4y^2 - 7x + 8y + 20 = 0 \)

### Step 2: Simplify the equations
We will simplify both equations by dividing them by their respective coefficients of \( x^2 \) and \( y^2 \).

For \( S_1 \):
\[ S_1 = 7(x^2 + y^2 - x + 2y + \frac{18}{7}) = 0 \implies x^2 + y^2 - x + 2y + \frac{18}{7} = 0 \]

For \( S_2 \):
\[ S_2 = 4(x^2 + y^2 - \frac{7}{4}x + 2y + 5) = 0 \implies x^2 + y^2 - \frac{7}{4}x + 2y + 5 = 0 \]

### Step 3: Find the equation of the radical axis
The equation of the radical axis is given by \( S_1 - S_2 = 0 \).

Subtract \( S_2 \) from \( S_1 \):
\[ S_1 - S_2 = \left( x^2 + y^2 - x + 2y + \frac{18}{7} \right) - \left( x^2 + y^2 - \frac{7}{4}x + 2y + 5 \right) = 0 \]

### Step 4: Simplify the equation
Cancelling \( x^2 \) and \( y^2 \):
\[ -x + \frac{7}{4}x + \frac{18}{7} - 5 = 0 \]

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

### Step 5: Find a common denominator and simplify
To combine the constants, convert 5 to a fraction with a denominator of 7:
\[ 5 = \frac{35}{7} \]
So:
\[ \frac{18}{7} - \frac{35}{7} = \frac{18 - 35}{7} = \frac{-17}{7} \]

Now, we have:
\[ \left(-1 + \frac{7}{4}\right)x - \frac{17}{7} = 0 \]

### Step 6: Simplify the coefficient of \( x \)
Calculate:
\[ -1 + \frac{7}{4} = -\frac{4}{4} + \frac{7}{4} = \frac{3}{4} \]
Thus:
\[ \frac{3}{4}x - \frac{17}{7} = 0 \]

### Step 7: Clear the fractions
Multiply through by 28 (the LCM of 4 and 7):
\[ 28 \left( \frac{3}{4}x \right) - 28 \left( \frac{17}{7} \right) = 0 \]
This simplifies to:
\[ 21x - 68 = 0 \]

### Conclusion
The equation of the radical axis is:
\[ 21x - 68 = 0 \]