The equation of the circle touching the axis of x at the origin and the line `4x - 3y + 24 = 0` is

Options

A. (0,12), 12
B. (0, -12), 12
C. (0,3), 3
D. (0, -3),3

Answer

To find the equation of the circle that touches the x-axis at the origin and the line \(4x - 3y + 24 = 0\), we can follow these steps:

### Step 1: Determine the center and radius of the circle
Since the circle touches the x-axis at the origin, the center of the circle must be at the point \((0, r)\), where \(r\) is the radius of the circle.

### Step 2: Find the distance from the center to the line
The distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by the formula:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \(4x - 3y + 24 = 0\), we have \(A = 4\), \(B = -3\), and \(C = 24\). The center of the circle is \((0, r)\).

### Step 3: Substitute the center into the distance formula
Substituting \((x_0, y_0) = (0, r)\) into the distance formula gives:
\[ d = \frac{|4(0) - 3(r) + 24|}{\sqrt{4^2 + (-3)^2}} = \frac{|-3r + 24|}{\sqrt{16 + 9}} = \frac{-3r + 24}{5} \]
Since the circle touches the line, this distance \(d\) must be equal to the radius \(r\):
\[ \frac{-3r + 24}{5} = r \]

### Step 4: Solve for \(r\)
To eliminate the fraction, multiply both sides by 5:
\[ -3r + 24 = 5r \]
Now, combine like terms:
\[ 24 = 5r + 3r \]
\[ 24 = 8r \]
Dividing both sides by 8 gives:
\[ r = 3 \]

### Step 5: Write the equation of the circle
Now that we have the radius \(r = 3\), the center of the circle is \((0, 3)\). The standard equation of a circle with center \((h, k)\) and radius \(r\) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = 0\), \(k = 3\), and \(r = 3\):
\[ (x - 0)^2 + (y - 3)^2 = 3^2 \]
This simplifies to:
\[ x^2 + (y - 3)^2 = 9 \]

### Final Answer
The equation of the circle is:
\[ x^2 + (y - 3)^2 = 9 \]