The circle whose centre is on the x-axis and the line `4x - 3y - 12 = 0` and whose radius is the distance between the lines `4x - 3y - 32= 0` and `4x - 3y - 12=0` has equation

Options

A. `x^(2)+y^(2)-6x-7=0`
B. `x^(2)+y^(2)-6x-8=0`
C. `x^(2)+y^(2)-6x-9=0`
D. none

Answer

To find the equation of the circle whose center lies on the x-axis, we will follow these steps:

### Step 1: Identify the center of the circle
The center of the circle is on the x-axis, which means its coordinates can be represented as \( (h, 0) \). We need to find the value of \( h \) using the line equation \( 4x - 3y - 12 = 0 \).

### Step 2: Substitute \( y = 0 \) into the line equation
Substituting \( y = 0 \) into the line equation:
\[ 4x - 3(0) - 12 = 0 \implies 4x - 12 = 0 \implies 4x = 12 \implies x = 3 \]
Thus, the center of the circle is \( (3, 0) \).

### Step 3: Find the radius of the circle
The radius of the circle is the distance between the two parallel lines \( 4x - 3y - 32 = 0 \) and \( 4x - 3y - 12 = 0 \).

### Step 4: Use the distance formula for parallel lines
The formula for the distance \( D \) between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by:
\[ D = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 4 \), \( B = -3 \), \( C_1 = -32 \), and \( C_2 = -12 \).

### Step 5: Calculate the distance
Substituting the values into the formula:
\[ D = \frac{|-12 - (-32)|}{\sqrt{4^2 + (-3)^2}} = \frac{|20|}{\sqrt{16 + 9}} = \frac{20}{\sqrt{25}} = \frac{20}{5} = 4 \]
Thus, the radius \( R \) of the circle is \( 4 \).

### Step 6: Write the equation of the circle
The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = 3 \), \( k = 0 \), and \( r = 4 \):
\[ (x - 3)^2 + (y - 0)^2 = 4^2 \]
This simplifies to:
\[ (x - 3)^2 + y^2 = 16 \]

### Step 7: Expand the equation
Expanding the equation:
\[ (x^2 - 6x + 9) + y^2 = 16 \]
Rearranging gives:
\[ x^2 + y^2 - 6x + 9 - 16 = 0 \implies x^2 + y^2 - 6x - 7 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 6x - 7 = 0 \]