The intercept made by the radical axis of the circles `x^2+y^2+6x-16=0` and `x^2+y^2-2x-6y-6=0` on x-axis is

Options

A. `3/4`
B. `5/4`
C. `1/4`
D. `7`

Answer

To find the intercept made by the radical axis of the circles given by the equations \( x^2 + y^2 + 6x - 16 = 0 \) and \( x^2 + y^2 - 2x - 6y - 6 = 0 \) on the x-axis, we will follow these steps:

### Step 1: Identify the equations of the circles
Let the first circle be represented as \( S_1 \):
\[ S_1: x^2 + y^2 + 6x - 16 = 0 \]

Let the second circle be represented as \( S_2 \):
\[ S_2: x^2 + y^2 - 2x - 6y - 6 = 0 \]

### Step 2: Find the radical axis
The radical axis of two circles \( S_1 \) and \( S_2 \) can be found by subtracting the equations:
\[ S_1 - S_2 = 0 \]

Substituting the equations:
\[ (x^2 + y^2 + 6x - 16) - (x^2 + y^2 - 2x - 6y - 6) = 0 \]

### Step 3: Simplify the equation
Now, simplify the expression:
\[ x^2 + y^2 + 6x - 16 - x^2 - y^2 + 2x + 6y + 6 = 0 \]
This simplifies to:
\[ (6x + 2x) + 6y - 16 + 6 = 0 \]
\[ 8x + 6y - 10 = 0 \]

### Step 4: Rearrange to find the equation of the line
Rearranging gives:
\[ 8x + 6y = 10 \]
Dividing the entire equation by 2:
\[ 4x + 3y = 5 \]

### Step 5: Find the intercept on the x-axis
To find the x-intercept, set \( y = 0 \):
\[ 4x + 3(0) = 5 \]
\[ 4x = 5 \implies x = \frac{5}{4} \]

### Conclusion
The intercept made by the radical axis of the given circles on the x-axis is:
\[ \frac{5}{4} \]